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

0,2

Comments

Diagonal of rational function 1/(1 - (x + y + x^2*y + x*y^2)). - Gheorghe Coserea, Aug 31 2018

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

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

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 *)

Prog

(PARI) a(n)=sum(k=0, n, binomial(2*k, k)*binomial(2*k, n-k));
(PARI) a(n)=polcoeff(1/sqrt(1-4*x*(1+x +x*O(x^n))^2), n, x);  /* Using the g.f.: */

Crossrefs

Cf. A137634, A137636, A137637, A137638, A073157.

Sequence in context: A221196 A137193 A006213 * A029706 A191644 A009640

Adjacent sequences: A137632 A137633 A137634 * A137636 A137637 A137638

Keyword

nonn

Author

Paul D. Hanna, Jan 31 2008

Status

approved