|
| |
|
|
A053266
|
|
Coefficients of the '5th-order' mock theta function Phi(q).
|
|
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 18, 20, 23.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
|
|
|
|
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))).
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.
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
