A121724 Generalized central binomial coefficients for k=2.

1, 1, 5, 9, 45, 97, 485, 1145, 5725, 14289, 71445, 185193, 925965, 2467137, 12335685, 33563481, 167817405, 464221105, 2321105525, 6507351113, 32536755565, 92236247841, 461181239205, 1319640776249, 6598203881245, 19031570387857, 95157851939285

Offset: 0

Author

Paul Barry, Aug 17 2006, Feb 28 2007

Keywords

• easy
• nonn

Comments

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)).

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000108, A009766, A053763, A120730.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (Sqrt(1-16*x^2)+2*x-1)/(2*x*(1-5*x)) )); // G. C. Greubel, Nov 07 2022

Mathematica

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

SageMath

def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
def A121724(n): return sum(4^(n-k)*A120730(n,k) for k in range(n+1))
[A121724(n) for n in range(51)] # G. C. Greubel, Nov 07 2022

Formula

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.

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).

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

a(n) = Sum_{k=0..floor(n/2)} A009766(n-k,k)*2^2k. - Philippe Deléham, Aug 18 2006

a(n) = Sum_{k=0..n} 4^(n-k)*A120730(n,k). - Philippe Deléham, Oct 16 2008

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

a(n) ~ (9+(-1)^n) * 2^(2*n+5/2) / (9 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 13 2014

Extensions

More terms from Vincenzo Librandi, Feb 15 2014