A387428 Expansion of 1/sqrt(1 - 12*x - 4*x^2).

1, 6, 56, 576, 6216, 68976, 779456, 8921856, 103098976, 1200177216, 14053176576, 165337030656, 1952904166656, 23143984805376, 275062611081216, 3277130498482176, 39128126836409856, 468065027383059456, 5608576020851019776, 67305503084237193216, 808786974964809035776

Offset: 0

Author

Seiichi Manyama, Aug 29 2025

Keywords

• nonn
• new

Crossrefs

Column k=3 of A386621.

Programs

PARI

a(n) = sum(k=0, n\2, 10^k*6^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k));

Formula

a(n) = Sum_{k=0..n} (3-i)^k * (3+i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit.

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

n*a(n) = 6*(2*n-1)*a(n-1) + 4*(n-1)*a(n-2) for n > 1.

a(n) = Sum_{k=0..floor(n/2)} 10^k * 6^(n-2*k) * binomial(n,2*k) * binomial(2*k,k).

a(n) = [x^n] (1+6*x+10*x^2)^n.

E.g.f.: exp(6*x) * BesselI(0, 2*sqrt(10)*x).

a(n) ~ 2^(n - 3/4) * (3 + sqrt(10))^(n + 1/2) / (5^(1/4) * sqrt(Pi*n)). - Vaclav Kotesovec, Aug 29 2025