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Search: a053256 -id:a053256
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A053263 Coefficients of the '5th-order' mock theta function chi_1(q). +10
50
1, 2, 2, 3, 3, 4, 4, 6, 5, 7, 8, 9, 9, 12, 12, 15, 15, 18, 19, 23, 23, 27, 30, 33, 34, 41, 42, 49, 51, 57, 61, 69, 72, 81, 87, 96, 100, 113, 119, 132, 140, 153, 163, 180, 188, 208, 221, 240, 253, 278, 294, 319, 339, 366, 388, 422, 443, 481, 510, 549, 580, 626, 662 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The rank of a partition is its largest part minus the number of parts.
Number of partitions of n such that 2*(least part) > greatest part. - Clark Kimberling, Feb 16 2014
Also the number of partitions of n with the same median as maximum. These are conjugate to the partitions described above. For minimum instead of maximum we have A361860. - Gus Wiseman, Apr 23 2023
REFERENCES
Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 20, 25
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from Seiichi Manyama)
George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.
FORMULA
G.f.: chi_1(q) = Sum_{n>=0} q^n/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))).
G.f.: chi_1(q) = 1 + Sum_{n>=0} q^(2n+1) (1+q^n)/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))).
a(n) is twice the number of partitions of 5n+3 with rank == 2 (mod 5) minus number with rank == 0 or 1 (mod 5).
a(n) - 1 is the number of partitions of n with unique smallest part and all other parts <= one plus twice the smallest part.
a(n) ~ sqrt(phi/2) * exp(Pi*sqrt(2*n/15)) / (5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 16 2019
EXAMPLE
From Gus Wiseman, Apr 20 2023: (Start)
The a(1) = 1 through a(8) = 6 partitions such that 2*(minimum) > (maximum):
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (32) (33) (43) (44)
(1111) (11111) (222) (322) (53)
(111111) (1111111) (332)
(2222)
(11111111)
The a(1) = 1 through a(8) = 6 partitions such that (median) = (maximum):
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (221) (33) (331) (44)
(1111) (11111) (222) (2221) (332)
(111111) (1111111) (2222)
(22211)
(11111111)
(End)
MATHEMATICA
1+Series[Sum[q^(2n+1)(1+q^n)/Product[1-q^k, {k, n+1, 2n+1}], {n, 0, 49}], {q, 0, 100}]
(* Also: *)
Table[Count[ IntegerPartitions[n], p_ /; 2 Min[p] > Max[p]], {n, 40}]
(* Clark Kimberling, Feb 16 2014 *)
nmax = 100; CoefficientList[Series[1 + Sum[x^(2*k+1)*(1+x^k) / Product[1-x^j, {j, k+1, 2*k+1}], {k, 0, Floor[nmax/2]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)
CROSSREFS
Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053262, A053264, A053265, A053266, A053267.
Cf. A047993, A240874, A361800, A361849 (ranks A361856), A361860.
A000041 counts integer partitions, strict A000009, odd-length A027193.
A359893 and A359901 count partitions by median.
Cf. A013580, A079309, A237753, A237757, A237800, A361848, A361858, A361859.
KEYWORD
nonn,easy
AUTHOR
Dean Hickerson, Dec 19 1999
STATUS
approved
A053261 Coefficients of the '5th-order' mock theta function psi_1(q). +10
17
1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 10, 10, 10, 11, 11, 12, 13, 13, 14, 15, 16, 16, 17, 18, 19, 20, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 34, 35, 37, 39, 40, 41, 44, 45, 47, 50, 51, 53, 56, 58, 60, 63, 65 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
COMMENTS
Number of partitions of n such that each part occurs at most twice and if k occurs as a part then all smaller positive integers occur.
Strictly unimodal compositions with rising range 1, 2, 3, ..., m where m is the largest part and distinct parts in the falling range (this follows trivially from the comment above). [Joerg Arndt, Mar 26 2014]
REFERENCES
Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 21, 22.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
William J. Keith, Partitions into parts simultaneously regular, distinct, and/or flat, Proceedings of CANT 2016; arXiv:1911.04755 [math.CO], 2019. Mentions this sequence.
George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.
FORMULA
G.f.: psi_1(q) = Sum_{n>=0} q^(n*(n+1)/2) * Product_{k=1..n} (1 + q^k).
a(n) ~ sqrt(phi) * exp(Pi*sqrt(n/15)) / (2*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i>n, 0, add(b(n-i*j, i+1), j=1..min(2, n/i))))
end:
a:= n-> b(n, 1):
seq(a(n), n=0..100); # Alois P. Heinz, Mar 26 2014
MATHEMATICA
Series[Sum[q^(n(n+1)/2) Product[1+q^k, {k, 1, n}], {n, 0, 13}], {q, 0, 100}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i > n, 0, Sum[b[n - i*j, i + 1], {j, 1, Min[2, n/i]}]]];
a[n_] := b[n, 1];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 09 2018, after Alois P. Heinz *)
nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/2) * Product[1+x^j, {j, 1, k}], {k, 0, Floor[Sqrt[2*nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)
PROG
(PARI) N = 66; x = 'x + O('x^N); gf = sum(n=0, N, x^(n*(n+1)/2) * prod(k=1, n, 1+x^k) ); v = Vec(gf) /* Joerg Arndt, Apr 21 2013 */
CROSSREFS
Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053262, A053263, A053264, A053265, A053266, A053267.
KEYWORD
nonn,easy
AUTHOR
Dean Hickerson, Dec 19 1999
STATUS
approved
A053266 Coefficients of the '5th-order' mock theta function Phi(q). +10
17
0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 9, 10, 12, 12, 14, 15, 17, 18, 20, 21, 25, 26, 29, 31, 35, 36, 41, 43, 48, 51, 56, 59, 66, 70, 76, 81, 89, 94, 103, 109, 119, 126, 137, 144, 158, 167, 180, 191, 207, 218, 236, 250, 269, 285, 306, 323, 349, 368 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
COMMENTS
In Ramanujan's lost notebook the generating function is denoted by phi(q) on pages 18 and 20, however on page 18 there is no minus one first term. - Michael Somos, Jul 07 2015
REFERENCES
G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, MR2952081, See p. 12, Equation (2.1.18) and also page 26 equation (2.4.8).
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 18, 20, 23.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
Dean Hickerson, A proof of the mock theta conjectures, Inventiones Mathematicae, 94 (1988) 639-660.
FORMULA
G.f.: -1 + Sum_{k>=0} q^(5k^2)/((1-q)(1-q^4)(1-q^6)(1-q^9)...(1-q^(5k+1))).
3*a(n) = A053262(n) + A259910(n) unless n=0. - Michael Somos, Jul 07 2015
a(n) ~ sqrt(phi/2) * exp(Pi*sqrt(2*n/15)) / (5^(3/4) * sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019
EXAMPLE
G.f. = x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 2*x^8 + 3*x^9 + ...
MATHEMATICA
a[ n_] := If[ n < 0, 0, With[ {m = Sqrt[1 + 24 n/5]}, SeriesCoefficient[ -1 + Sum[ (-1)^k x^(5 k (3 k + 1)/2) / (1 - x^(5 k + 1)), {k, Quotient[m + 1, -6], Quotient[m - 1, 6]}] / QPochhammer[ x^5], {x, 0, n}]]]; (* Michael Somos, Jul 07 2015 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n\5), x^(5*k^2) / prod(i=1, 5*k+1, 1 - if( i%5==1 || i%5==4, x^i), 1 + x * O(x^(n - 5*k^2)))) - 1, n))}; /* Michael Somos, Jul 07 2015 */
(PARI) {a(n) = my(A, m); if( n<0, 0, m = sqrtint(1 + 24*n\5); A = x * O(x^n); polcoeff( sum(k=(m + 1)\-6, (m - 1)\6, (-1)^k * x^(5*k*(3*k + 1)/2) / (1 - x^(5*k + 1)), A) / eta(x^5 + A) - 1, n))}; /* Michael Somos, Jul 07 2015 */
CROSSREFS
Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053265, A053267.
Cf. A259910.
KEYWORD
nonn,easy
AUTHOR
Dean Hickerson, Dec 19 1999
STATUS
approved
A053258 Coefficients of the '5th-order' mock theta function phi_0(q). +10
15
1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 4, 4, 3, 4, 4, 3, 4, 4, 5, 4, 4, 5, 5, 5, 5, 6, 6, 5, 5, 6, 6, 6, 6, 7, 7, 7, 6, 7, 8, 7, 8, 8, 9, 9, 8, 9, 10, 9, 9, 10, 11, 10, 10, 11, 11, 11, 11, 12 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,18
REFERENCES
Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 22, 23, 25.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.
FORMULA
G.f.: phi_0(q) = Sum_{n>=0} q^n^2 (1+q)(1+q^3)...(1+q^(2n-1)).
a(n) is the number of partitions of n into odd parts such that each occurs at most twice and if k occurs as a part then all smaller positive odd numbers occur.
a(n) ~ sqrt(phi) * exp(Pi*sqrt(n/30)) / (2*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019
MATHEMATICA
Series[Sum[q^n^2 Product[1+q^(2k-1), {k, 1, n}], {n, 0, 10}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(k^2)*Product[1+x^(2*j-1), {j, 1, k}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 11 2019 *)
CROSSREFS
Other '5th-order' mock theta functions are at A053256, A053257, A053259, A053260, A053261, A053262, A053263, A053264, A053265, A053266, A053267.
KEYWORD
nonn,easy
AUTHOR
Dean Hickerson, Dec 19 1999
STATUS
approved
A053262 Coefficients of the 5th-order mock theta function chi_0(q). +10
15
1, 1, 1, 2, 1, 3, 2, 3, 3, 5, 3, 6, 5, 7, 7, 9, 7, 12, 11, 13, 13, 17, 15, 21, 20, 24, 24, 29, 28, 36, 35, 40, 42, 50, 48, 58, 58, 67, 70, 80, 79, 93, 95, 106, 111, 125, 127, 145, 149, 166, 172, 191, 196, 222, 229, 250, 262, 289, 298, 330, 343, 373, 391, 427, 442, 486 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
The rank of a partition is its largest part minus the number of parts.
REFERENCES
Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 20, 23, 25.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (corrected and extended previous b-file from G. C. Greubel)
George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.
FORMULA
G.f.: chi_0(q) = Sum_{n>=0} q^n/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n))).
G.f.: chi_0(q) = 1 + Sum_{n>=0} q^(2n+1)/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))).
a(n) is the number of partitions of 5n with rank == 1 (mod 5) minus number with rank == 0 (mod 5).
a(n) is the number of partitions of n with unique smallest part and all other parts <= twice the smallest part.
a(n) is the number of partitions where the largest part is odd and all other parts are greater than half of the largest part. - N. Sato, Jan 21 2010
a(n) ~ exp(Pi*sqrt(2*n/15)) / sqrt((5 + sqrt(5))*n). - Vaclav Kotesovec, Jun 12 2019
MATHEMATICA
1+Series[Sum[q^(2n+1)/Product[1-q^k, {k, n+1, 2n+1}], {n, 0, 49}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[1 + Sum[x^(2*k+1)/Product[1-x^j, {j, k+1, 2*k+1}], {k, 0, Floor[nmax/2]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)
CROSSREFS
Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053263, A053264, A053265, A053266, A053267.
KEYWORD
nonn,easy
AUTHOR
Dean Hickerson, Dec 19 1999
STATUS
approved
A053264 Coefficients of the '5th-order' mock theta function F_0(q). +10
14
1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6, 6, 7, 8, 8, 10, 11, 11, 13, 14, 15, 17, 18, 19, 22, 24, 25, 28, 30, 32, 36, 39, 41, 45, 49, 52, 57, 61, 65, 71, 76, 81, 88, 94, 100, 109, 116, 123, 133, 142, 151, 163, 174, 184, 198, 211, 224, 240, 255, 271, 290 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,9
REFERENCES
Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 20, 22, 23, 25.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.
FORMULA
G.f.: F_0(q) = Sum_{n>=0} q^(2n^2)/((1-q)(1-q^3)...(1-q^(2n-1))).
a(n) is the number of partitions of n into odd parts, each of which occurs at least twice, such that if k occurs then all smaller positive odd numbers occur.
a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(3/2)*5^(1/4)*sqrt(phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019
MATHEMATICA
Series[Sum[q^(2n^2)/Product[1-q^(2k+1), {k, 0, n-1}], {n, 0, 7}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(2*k^2) / Product[1-x^(2*j+1), {j, 0, k-1}], {k, 0, Floor[Sqrt[nmax/2]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)
CROSSREFS
Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053262, A053263, A053265, A053266, A053267.
KEYWORD
nonn,easy
AUTHOR
Dean Hickerson, Dec 19 1999
STATUS
approved
A053267 Coefficients of the '5th-order' mock theta function Psi(q). +10
14
0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 4, 3, 4, 4, 5, 5, 7, 6, 8, 8, 9, 9, 12, 11, 14, 14, 16, 16, 20, 19, 23, 24, 27, 27, 32, 32, 37, 38, 43, 44, 51, 51, 58, 61, 67, 69, 78, 80, 89, 93, 102, 106, 118, 121, 134, 140, 153, 159, 175, 181, 198, 207, 224, 234, 256, 265, 288 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,9
REFERENCES
Dean Hickerson, A proof of the mock theta conjectures, Inventiones Mathematicae, 94 (1988) 639-660
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 18, 20
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
FORMULA
G.f.: Psi(q) = -1 + Sum_{n>=0} q^(5n^2)/((1-q^2)(1-q^3)(1-q^7)(1-q^8)...(1-q^(5n+2))).
a(n) ~ exp(Pi*sqrt(2*n/15)) / (5^(3/4)*sqrt(2*phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019
MATHEMATICA
Series[Sum[q^(5n^2)/Product[1-q^Abs[5k+2], {k, -n, n}], {n, 0, 4}], {q, 0, 100}]-1
nmax = 100; CoefficientList[Series[-1 + Sum[x^(5*k^2)/ Product[1-x^Abs[5*j+2], {j, -k, k}], {k, 0, Floor[Sqrt[nmax/5]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)
CROSSREFS
Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053265, A053266.
KEYWORD
nonn,easy
AUTHOR
Dean Hickerson, Dec 19 1999
STATUS
approved
A053260 Coefficients of the '5th-order' mock theta function psi_0(q). +10
13
0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 3, 3, 3, 3, 4, 4, 4, 5, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 9, 8, 9, 10, 9, 11, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 17, 18, 19, 19, 21, 22, 22, 24, 25, 25, 27, 28, 29, 30, 32, 32, 34, 36, 36, 39, 40, 41, 44, 45, 46 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,14
COMMENTS
Number of partitions of n such that each part occurs at most twice, the largest part is unique and if k occurs as a part then all smaller positive integers occur.
Strongly unimodal compositions with first part 1 and each up-step is by at most 1 (left-smoothness); with this interpretation one should set a(0)=1; see example. Replacing "strongly" by "weakly" in the condition gives A001524. Dropping the requirement of unimodality gives A005169. [Joerg Arndt, Dec 09 2012]
REFERENCES
Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 21, 22.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000
George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.
FORMULA
G.f.: psi_0(q) = Sum_{n>=0} q^((n+1)*(n+2)/2) * (1+q)*(1+q^2)*...*(1+q^n).
a(n) ~ exp(Pi*sqrt(n/15)) / (2*5^(1/4)*sqrt(phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019
EXAMPLE
From Joerg Arndt, Dec 09 2012: (Start)
The a(42)=8 strongly unimodal left-smooth compositions are
[ #] composition
[1] [ 1 2 3 4 5 6 7 5 4 3 2 ]
[2] [ 1 2 3 4 5 6 7 6 4 3 1 ]
[3] [ 1 2 3 4 5 6 7 6 5 2 1 ]
[4] [ 1 2 3 4 5 6 7 6 5 3 ]
[5] [ 1 2 3 4 5 6 7 8 3 2 1 ]
[6] [ 1 2 3 4 5 6 7 8 4 2 ]
[7] [ 1 2 3 4 5 6 7 8 5 1 ]
[8] [ 1 2 3 4 5 6 7 8 6 ]
(End)
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+`if`(i>n, 0, b(n-i, i-1))))
end:
a:= proc(n) local h, k, m, r;
m, r:= floor((sqrt(n*8+1)-1)/2), 0;
for k from m by -1 do h:= k*(k+1);
if h<=n then break fi;
r:= r+b(n-h/2, k-1)
od: r
end:
seq(a(n), n=0..100); # Alois P. Heinz, Aug 02 2013
MATHEMATICA
Series[Sum[q^((n+1)(n+2)/2) Product[1+q^k, {k, 1, n}], {n, 0, 12}], {q, 0, 100}]
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i-1] ] ]]; a[n_] := Module[{h, k, m, r}, {m, r} = {Floor[(Sqrt[n*8+1]-1)/2], 0}; For[k = m, True, k--, h = k*(k+1); If[h <= n, Break[]]; r = r + b[n-h/2, k-1]]; r]; Table[ a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 09 2015, after Alois P. Heinz *)
PROG
(PARI)
N = 66; x = 'x + O('x^N);
gf = sum(n=1, N, x^(n*(n+1)/2) * prod(k=1, n-1, 1+x^k) ) + 'c0;
v = Vec(gf); v[1]-='c0; v
/* Joerg Arndt, Apr 21 2013 */
CROSSREFS
Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053261, A053262, A053263, A053264, A053265, A053266, A053267.
KEYWORD
nonn,easy
AUTHOR
Dean Hickerson, Dec 19 1999
STATUS
approved
A053265 Coefficients of the '5th-order' mock theta function F_1(q). +10
13
1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 18, 20, 21, 24, 26, 28, 31, 34, 37, 40, 44, 47, 51, 56, 60, 65, 71, 76, 82, 89, 95, 103, 111, 119, 128, 138, 148, 158, 171, 182, 195, 210, 223, 239, 256, 273, 292, 312, 332, 354, 378, 402, 428 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
REFERENCES
Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 20, 22, 25.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.
FORMULA
G.f.: F_1(q) = Sum_{n>=0} q^(2n(n+1))/((1-q)(1-q^3)...(1-q^(2n+1))).
a(n) ~ sqrt(phi) * exp(Pi*sqrt(2*n/15)) / (2^(3/2)*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019
MATHEMATICA
Series[Sum[q^(2n(n+1))/Product[1-q^(2k+1), {k, 0, n}], {n, 0, 6}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(2*k*(k+1)) / Product[1-x^(2*j+1), {j, 0, k}], {k, 0, Floor[Sqrt[nmax/2]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)
CROSSREFS
Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053266, A053267.
KEYWORD
nonn,easy
AUTHOR
Dean Hickerson, Dec 19 1999
STATUS
approved
A053257 Coefficients of the '5th-order' mock theta function f_1(q). +10
12
1, 0, 1, -1, 1, -1, 2, -2, 1, -1, 2, -2, 2, -2, 2, -3, 3, -2, 3, -4, 4, -4, 4, -5, 5, -4, 5, -6, 6, -6, 7, -8, 7, -7, 8, -9, 10, -9, 10, -12, 11, -11, 13, -14, 14, -15, 16, -17, 17, -16, 19, -21, 20, -21, 23, -25, 25, -25, 27, -29, 30, -30, 32, -35, 35, -35, 39, -41, 41, -43, 45, -49, 50, -49, 53, -57, 58, -59, 63, -67, 68 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
REFERENCES
Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 22.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
Dean Hickerson, A proof of the mock theta conjectures, Inventiones Mathematicae, 94 (1988) 639-660.
George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.
FORMULA
G.f.: f_1(q) = Sum_{n>=0} q^(n^2+n)/((1+q)(1+q^2)...(1+q^n)).
Consider partitions of n into parts differing by at least 2 and with smallest part at least 2. a(n) is the number of them with largest part even minus number with largest part odd.
a(n) ~ (-1)^n * sqrt(phi) * exp(Pi*sqrt(n/15)) / (2*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 15 2019
MATHEMATICA
Series[Sum[q^(n^2+n)/Product[1+q^k, {k, 1, n}], {n, 0, 9}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(k^2+k) / Product[1+x^j, {j, 1, k}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 15 2019 *)
CROSSREFS
Other '5th-order' mock theta functions are at A053256, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053265, A053266, A053267.
KEYWORD
sign,easy
AUTHOR
Dean Hickerson, Dec 19 1999
STATUS
approved
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