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A121724 Generalized central binomial coefficients for k=2. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Hankel transform is 4^binomial(n+1,2) = A053763(n+1). Case k=2 of T(n,k) = (1/Pi)*2*k^2*(2k)^n*Integral_{x=-1..1} x^n*sqrt(1-x^2)/(1+k^2-2kx) 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+2x+5x^2).

LINKS

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

FORMULA

G.f.: (sqrt(1-16x^2)+2x-1)/(2x(1-5x)) = c(4x^2)/(1-x*c(4x^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(2n-2k+j,n-2k+j)*(-1)^(n-2k+j)*2^j*5^(k-j).

a(n) = (1/Pi)*8*4^n*Integral_{x=-1..1} x^n*sqrt(1-x^2)/(5-4x) 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) = 0. - 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

MATHEMATICA

CoefficientList[Series[(Sqrt[1-16*x^2]+2*x-1)/(2*x*(1-5*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

CROSSREFS

Cf. A000108, A009766, A053763, A120730.

Sequence in context: A304127 A220518 A145031 * A149496 A149497 A149498

Adjacent sequences:  A121721 A121722 A121723 * A121725 A121726 A121727

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Aug 17 2006, Feb 28 2007

EXTENSIONS

More terms from Vincenzo Librandi, Feb 15 2014

STATUS

approved

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Last modified January 16 16:32 EST 2022. Contains 350376 sequences. (Running on oeis4.)