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A102426
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Triangle read by rows giving coefficients of polynomials defined by F(0,x)=0, F(1,x)=1, F(n,x) = F(n-1,x) + x*F(n-2,x).
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21
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0, 1, 1, 1, 1, 2, 1, 1, 3, 1, 3, 4, 1, 1, 6, 5, 1, 4, 10, 6, 1, 1, 10, 15, 7, 1, 5, 20, 21, 8, 1, 1, 15, 35, 28, 9, 1, 6, 35, 56, 36, 10, 1, 1, 21, 70, 84, 45, 11, 1, 7, 56, 126, 120, 55, 12, 1, 1, 28, 126, 210, 165, 66, 13, 1, 8, 84, 252, 330, 220, 78, 14, 1, 1, 36, 210, 462, 495, 286, 91, 15, 1
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OFFSET
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0,6
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COMMENTS
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F(n) + 2x * F(n-1) gives Lucas polynomials (cf. A034807). - Maxim Krikun (krikun(AT)iecn.u-nancy.fr), Jun 24 2007
After the initial 0, these are the nonzero coefficients of the Fibonacci polynomials; see the Mathematica section. - Clark Kimberling, Oct 10 2013
Aside from signs and index shift, the coefficients of the characteristic polynomial of the Coxeter adjacency matrix for the Coxeter group A_n related to the Chebyshev polynomial of the second kind (cf. Damianou link p. 19). - Tom Copeland, Oct 11 2014
Aside from the initial zeros, these are the antidiagonals read from bottom to top of the numerical coefficients of the Maurer-Cartan form matrix of the Leibniz group L^(n)(1,1) presented on p. 9 of the Olver paper, which is generated as exp[c. * M] with (c.)^n = c_n and M the Lie infinitesimal generator A218272. Reverse of A011973. - Tom Copeland, Jul 02 2018
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REFERENCES
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Dominique Foata and Guo-Niu Han, Multivariable tangent and secant q-derivative polynomials, Manuscript, Mar 21 2012.
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LINKS
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FORMULA
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Alternatively, as n is even or odd: T(n-2, k) + T(n-1, k-1) = T(n, k), T(n-2, k) + T(n-1, k) = T(n, k)
T(n, k) = binomial(floor(n/2)+k, floor((n-1)/2-k) ). - Paul Barry, Jun 22 2005
Beginning with the second polynomial in the example and offset=0, P(n,t)= Sum_{j=0..n}, binomial(n-j,j)*x^j with the convention that 1/k! is zero for k=-1,-2,..., i.e., 1/k! = lim_{c->0} 1/(k+c)!. - Tom Copeland, Oct 11 2014
O.g.f.: (x + x^2 - x^3) / (1 - (2+t)*x^2 + x^4) = (x^2 (even part) + x*(1-x^2) (odd)) / (1 - (2+t)*x^2 + x^4).
Recursion relations:
A) p(n,t) = p(n-1,t) + p(n-2,t) for n=2,4,6,8,...
B) p(n,t) = t*p(n-1,t) + p(n-2,t) for n=3,5,7,...
C) a(n,k) = a(n-2,k) + a(n-1,k) for n=4,6,8,...
D) a(n,k) = a(n-2,k) + a(n-1,k-1) for n=3,5,7,...
Relation A generalized to MV(n,t;r) = P(2n+1,t) + r R(2n,t) for n=1,2,3,... (cf. A078812 and A085478) is the generating relation on p. 229 of Andre-Jeannine for the generalized Morgan-Voyce polynomials, e.g., MV(2,t;r) = p(5,t) + r*p(4,t) = (1 + 3t + t^2) + r*(2 + t) = (1 + 2r) + (3 + r)*t + t^2, so P(n,t) = MV(n-4,t;1) for n=4,6,8,... .
The even and odd polynomials are also presented in Trzaska and Ferri.
Dropping the initial 0 and re-indexing with initial m=0 gives the row polynomials Fb(m,t) = p(n+1,t) below with o.g.f. G(t,x)/x, starting with Fb(0,t) = 1, Fb(1,t) = 1, Fb(2,t) = 1 + t, and Fb(3,t) = 2 + t.
The o.g.f. x/G(x,t) = (1 - (2+t)*x^2 + x^4) / (1 + x - x^2) then generates a sequence of polynomials IFb(t) such that the convolution Sum_{k=0..n} IFb(n-k,t) Fb(k,t) vanishes for n>1 and is one for n=0. These linear polynomials have the basic Fibonacci numbers A000045 as an overall factor:
IFb(0,t) = 1
IFb(1,t) = -1
IFb(2,t) = -t
IFb(3,t) = -1 (1-t)
IFb(4,t) = 2 (1-t)
IFb(5,t) = -3 (1-t)
IFb(6,t) = 5 (1-t)
IFb(7,t) = -8 (1-t)
IFb(8,t) = 13 (1-t)
... .
(End)
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EXAMPLE
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The first few polynomials are:
0
1
1
x + 1
2*x + 1
x^2 + 3*x + 1
3*x^2 + 4*x + 1
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[n]:
0: 0
1: 1
2: 1
3: 1 1
4: 2 1
5: 1 3 1
6: 3 4 1
7: 1 6 5 1
8: 4 10 6 1
9: 1 10 15 7 1
10: 5 20 21 8 1
11: 1 15 35 28 9 1
12: 6 35 56 36 10 1
13: 1 21 70 84 45 11 1
(End)
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MATHEMATICA
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Join[{0}, Table[ Select[ CoefficientList[ Fibonacci[n, x], x], 0 < # &], {n, 0, 17}]//Flatten] (* Clark Kimberling, Oct 10 2013 and slightly modified by Robert G. Wilson v, May 03 2017 *)
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PROG
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(Magma) [0] cat [Binomial(Floor(n/2)+k, Floor((n-1)/2-k) ): k in [0..Floor((n-1)/2)], n in [0..17]]; // G. C. Greubel, Oct 13 2019
(PARI) F(n) = if (n==0, 0, if (n==1, 1, F(n-1) + x*F(n-2)));
tabf(nn) = for (n=0, nn, print(Vec(F(n)))); \\ Michel Marcus, Feb 10 2020
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CROSSREFS
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KEYWORD
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easy,nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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