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
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
Programs
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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)}
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