A137636 a(n) = Sum_{k=0..n} C(2k+1,k)*C(2k+1,n-k) ; equals row 1 of square array A137634; also equals the convolution of A137635 and A073157.

1, 4, 19, 94, 474, 2431, 12609, 65972, 347524, 1840680, 9792986, 52296799, 280163091, 1504969409, 8103433329, 43722788132, 236340999038, 1279602656590, 6938126362948, 37668424608552, 204751452911832, 1114151447523038

Offset

0,2

Links

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

Formula

G.f.: A(x) = R(x)*G(x), where R(x) = 1/sqrt(1-4x(1+x)^2) is the g.f. of A137635 and G(x) = (1-sqrt(1-4x(1+x)^2))/(2x(1+x)) is the g.f. of A073157.

D-finite with recurrence (n+1)*a(n) +(-3*n-1)*a(n-1) +2*(-6*n-1)*a(n-2) +2*(-6*n+1)*a(n-3) +2*(-2*n+1)*a(n-4)=0. - R. J. Mathar, Jun 23 2023

a(n) ~ sqrt((172 + (86*(78905 - 519*sqrt(129)))^(1/3) + (86*(78905 + 519*sqrt(129)))^(1/3))/129) * ((4 + (262 - 6*sqrt(129))^(1/3) + (2*(131 + 3*sqrt(129)))^(1/3))/3)^n / sqrt(Pi*n). - Vaclav Kotesovec, Nov 25 2023

Prog

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

Crossrefs

Cf. A137634, A137635, A137637, A137638, A073157.

Sequence in context: A122369 A005978 A083065 * A027618 A278678 A020060

Adjacent sequences: A137633 A137634 A137635 * A137637 A137638 A137639

Keyword

nonn

Author

Paul D. Hanna, Jan 31 2008

Status

approved