OFFSET
0,2
COMMENTS
Because (x*c(x))^n + (1/c(x))^n = L(n,-x)= Sum_{k=0..floor(n/2)} A034807(n,k)*(-x)^k the sequence starts with the Lucas polynomial L(10,-x) (of degree 5).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1671
FORMULA
O.g.f.: 1/c(x)^10 = P(11, x) - x*P(10, x)*c(x) with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers) and the polynomials P(n, x) defined in A115139. Here P(11, x) = 1 - 9*x + 28*x^2 - 35*x^3 + 15*x^4 - x^5 and P(10, x) = 1 - 8*x + 21*x^2 - 20*x^3 + 5*x^4.
a(n) = -C10(n-10), n >= 10, with C10(n) = (n+4) (eighth convolution of Catalan numbers). a(0)=1, a(1)=-10, a(2)=35, a(3)=-50, a(4)=25, a(5)=-2, a(6)=a(7)=a(8)=a(9)=0. [1, -10, 35, -50, 25, -2] is row n=10 of signed A034807 (signed Lucas polynomials).
a(n) ~ -5 * 2^(2*n - 10) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 13 2019
MATHEMATICA
CoefficientList[Series[(1-10*x+35*x^2-50*x^3+25*x^4-2*x^5 +(1-8*x+21*x^2 -20*x^3+5*x^4)*Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)
PROG
(PARI) my(x='x+O('x^30)); Vec((1-10*x+35*x^2-50*x^3+25*x^4-2*x^5 +(1-8*x +21*x^2 -20*x^3+5*x^4)*sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-10*x+35*x^2-50*x^3+25*x^4-2*x^5 +(1-8*x+21*x^2 -20*x^3+5*x^4 )*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019
(Sage) ((1-10*x+35*x^2-50*x^3+25*x^4-2*x^5 +(1-8*x+21*x^2 -20*x^3+5*x^4) *sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Wolfdieter Lang, Jan 13 2006
STATUS
approved