A036911 a(n) = (binomial(4*n, 2*n) + (-1)^n*binomial(2*n, n)^2)/2.
1, 1, 53, 262, 8885, 60626, 1778966, 14168988, 383358645, 3355615450, 85990654178, 803232328548, 19780031677718, 193873026294052, 4629016098160220, 47101568276955512, 1096960888092571317, 11503661742608944170, 262435310495071434602
Offset: 0
Keywords
References
- Identity (3.71) in H. W. Gould, Combinatorial Identities, Morgantown, 1972, page 30.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..825
Programs
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Magma
[(1/2)*((2*n+1)*Catalan(2*n) + (-1)^n*(n+1)^2*Catalan(n)^2): n in [0..30]]; // G. C. Greubel, Jun 22 2022
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Mathematica
Table[(Binomial[4n,2n]+(-1)^n Binomial[2n,n]^2)/2,{n,0,20}] (* Harvey P. Dale, May 22 2013 *) -
SageMath
b=binomial; [(1/2)*(b(4*n, 2*n) + (-1)^n*b(2*n, n)^2) for n in (0..30)] # G. C. Greubel, Jun 22 2022
Formula
a(n) = (1/2)*(binomial(4*n, 2*n) + (-1)^n*binomial(2*n, n)^2).
From G. C. Greubel, Jun 22 2022: (Start)
a(n) = Sum_{k=0..n} binomial(2*n, 2*k)^2.
G.f.: (1/2)*( sqrt(1 + sqrt(1-16*x))/(sqrt(2)*sqrt(1-16*x)) + (2/Pi)*EllipticK(-16*x) ). (End)