A137635 a(n) = Sum_{k=0..n} C(2k,k)*C(2k,n-k); equals row 0 of square array A137634.
1, 2, 10, 46, 226, 1136, 5810, 30080, 157162, 826992, 4376408, 23267332, 124179570, 664919780, 3570265000, 19216805476, 103652442922, 560127574340, 3031887311256, 16435458039076, 89213101943000, 484839755040768, 2637805800869740, 14365506336197816
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
CoefficientList[Series[1/Sqrt[1 - 4*x*(1 + x)^2],{x,0,50}],x] (* Stefano Spezia, Sep 01 2018 *) Table[Sum[Binomial[2k,k]Binomial[2k,n-k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Dec 31 2018 *) a[n_]:=Binomial[2n, n]HypergeometricPFQ[{(1-2*n)/3, 2(1-n)/3, -2n/3}, {1/2-n, 1/2-n}, -3^3/2^4]; Array[a,24,0] (* Stefano Spezia, Jul 11 2024 *) -
PARI
a(n)=sum(k=0,n,binomial(2*k,k)*binomial(2*k,n-k));
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PARI
a(n)=polcoeff(1/sqrt(1-4*x*(1+x +x*O(x^n))^2),n,x); /* Using the g.f.: */
Formula
G.f.: A(x) = 1/sqrt(1 - 4x(1+x)^2).
D-finite with recurrence: n*a(n) +2*(-2*n+1)*a(n-1) +8*(-n+1)*a(n-2) +2*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Jan 14 2020
a(n) = binomial(2*n, n)*hypergeom([(1-2*n)/3, 2*(1-n)/3, -2*n/3], [1/2-n, 1/2-n], -3^3/2^4). - Stefano Spezia, Jul 11 2024
Comments