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Expansion of c(5x^2)/(1-x*c(5x^2)), where c(x) is the g.f. of A000108.
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%I #16 Nov 08 2022 01:47:08

%S 1,1,6,11,66,146,876,2131,12786,32966,197796,530526,3183156,8786436,

%T 52718616,148733571,892401426,2561439806,15368638836,44731364266,

%U 268388185596,790211926076,4741271556456,14095578557486

%N Expansion of c(5x^2)/(1-x*c(5x^2)), where c(x) is the g.f. of A000108.

%C Hankel transform is 5^C(n+1,2).

%C Reversion of x*(1+x)/(1+2*x+6*x^2).

%H G. C. Greubel, <a href="/A128387/b128387.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (sqrt(1-20*x^2) + 2*x - 1)/(2*x*(1-6*x)).

%F a(n) = (1/(n+1))*Sum_{k=0..n+1} Sum_{j=0..k} C(n,k)*C(k,j)*C(2*n-2*k+j, n-2*k+j)*(-1)^(n+j)*2^j*6^(k-j).

%F a(n) = Sum_{k=0..floor(n/2)} C(n,n-k)*(n-2*k+1)*5^k/(n-k+1).

%F a(n) = Sum_{k=0..floor(n/2)} A009766(n-k,k)*5^k.

%F a(n) = Sum_{k=0..n} 5^k*A120730(n,n-k). - _Philippe Deléham_, Mar 03 2007

%F (n+1)*a(n) = 6*(n+1)*a(n-1) + 20*(n-2)*a(n-2) - 120*(n-2)*a(n-3). - _R. J. Mathar_, Nov 14 2011

%t A120730[n_, k_]:= If[n>2*k, 0, Binomial[n, k]*(2*k-n+1)/(k+1)];

%t A126387[n_]:= Sum[5^k*A120730[n, n-k], {k,0,n}];

%t Table[A126387[n], {n, 0, 50}] (* _G. C. Greubel_, Nov 07 2022 *)

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 50); Coefficients(R!( (Sqrt(1-20*x^2)+2*x-1)/(2*x*(1-6*x)) )); // _G. C. Greubel_, Nov 07 2022

%o (SageMath)

%o def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)

%o def A126387(n): return sum(5^k*A120730(n,n-k) for k in range(n+1))

%o [A126387(n) for n in range(51)] # _G. C. Greubel_, Nov 07 2022

%Y Cf. A000108, A009766, A120730, A126386.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Feb 28 2007