%I #17 Jul 31 2024 06:17:30
%S 1,-10,35,-50,25,-2,0,0,0,0,-1,-10,-65,-350,-1700,-7752,-33915,
%T -144210,-600875,-2466750,-10015005,-40320150,-161280600,-641886000,
%U -2544619500,-10056336264,-39645171810,-155989499540,-612815891050,-2404551645100,-9425842448792
%N Tenth convolution of A115140.
%C 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).
%H Seiichi Manyama, <a href="/A115149/b115149.txt">Table of n, a(n) for n = 0..1671</a>
%F 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.
%F 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).
%F a(n) ~ -5 * 2^(2*n - 10) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 13 2019
%t 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 *)
%o (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
%o (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
%o (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
%Y Cf. A003519, A115139 - A115148.
%K sign,easy
%O 0,2
%A _Wolfdieter Lang_, Jan 13 2006