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Generalized central binomial coefficients for k=2.
6

%I #23 Nov 08 2022 01:47:12

%S 1,1,5,9,45,97,485,1145,5725,14289,71445,185193,925965,2467137,

%T 12335685,33563481,167817405,464221105,2321105525,6507351113,

%U 32536755565,92236247841,461181239205,1319640776249,6598203881245,19031570387857,95157851939285

%N Generalized central binomial coefficients for k=2.

%C Hankel transform is 4^binomial(n+1,2) = A053763(n+1). Case k=2 of T(n,k) = (1/Pi)*2*k^2*(2*k)^n*Integral_{x=-1..1} x^n*sqrt(1-x^2)/(1+k^2-2*k*x) dx. T(n,k) has Hankel transform (k^2)^binomial(n+1,2). k=1 corresponds to C(n,floor(n/2)).

%C Series reversion of x*(1+x)/(1+2*x+5*x^2).

%H Vincenzo Librandi, <a href="/A121724/b121724.txt">Table of n, a(n) for n = 0..200</a>

%F G.f.: (sqrt(1-16*x^2) + 2*x - 1)/(2*x*(1-5*x)) = c(4*x^2)/(1-x*c(4*x^2)), c(x) the g.f. of A000108.

%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-2*k+j)*2^j*5^(k-j).

%F a(n) = (1/Pi)*8*4^n*Integral_{x=-1..1} x^n*sqrt(1-x^2)/(5-4*x) dx.

%F a(n) = Sum_{k=0..floor(n/2)} A009766(n-k,k)*2^2k. - _Philippe Deléham_, Aug 18 2006

%F a(n) = Sum_{k=0..n} 4^(n-k)*A120730(n,k). - _Philippe Deléham_, Oct 16 2008

%F Conjecture: (n+1)*a(n) = 5*(n+1)*a(n-1) + 16*(n-2)*a(n-2) - 80*(n-2)*a(n-3). - _R. J. Mathar_, Nov 26 2012

%F a(n) ~ (9+(-1)^n) * 2^(2*n+5/2) / (9 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 13 2014

%t CoefficientList[Series[(Sqrt[1-16*x^2]+2*x-1)/(2*x*(1-5*x)), {x,0,40}], x] (* _Vaclav Kotesovec_, Feb 13 2014 *)

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (Sqrt(1-16*x^2)+2*x-1)/(2*x*(1-5*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 A121724(n): return sum(4^(n-k)*A120730(n,k) for k in range(n+1))

%o [A121724(n) for n in range(51)] # _G. C. Greubel_, Nov 07 2022

%Y Cf. A000108, A009766, A053763, A120730.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Aug 17 2006, Feb 28 2007

%E More terms from _Vincenzo Librandi_, Feb 15 2014