A386940 - OEIS

A386940

a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(2*n-k-1,n-k).

1, 3, 13, 60, 285, 1378, 6748, 33372, 166365, 834900, 4213638, 21368724, 108820764, 556184580, 2851679620, 14661848560, 75568345821, 390330333402, 2020046912260, 10472193542100, 54373036935910, 282704274266040, 1471722678992700, 7670327017789800, 40017679829372700

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OFFSET

0,2

LINKS

Table of n, a(n) for n=0..24.

FORMULA

a(n) = [x^n] 1/(sqrt(1-4*x) * (1-x)^n).

G.f.: (1+sqrt(1-4*x))/sqrt( 4 * (1-4*x) * (2*sqrt(1-4*x)-1) ).

a(n) = Sum_{k=0..n} 3^(n-k) * binomial(2*n-1/2,k) * binomial(n-k-1/2,n-k) = Sum_{k=0..n} (3/4)^k * binomial(2*k,k) * binomial(2*n-1/2,n-k).

a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(2*n-1/2,k) * binomial(2*n-k-1,n-k).

PROG

(PARI) a(n) = sum(k=0, n, binomial(2*k, k)*binomial(2*n-k-1, n-k));