A360318 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

1, 2, 12, 74, 466, 2982, 19320, 126390, 833220, 5527190, 36852052, 246751854, 1658106394, 11176100138, 75528743352, 511600414554, 3472363279170, 23609924743590, 160788499672020, 1096566516149790, 7488135911236806, 51193972101241362, 350368409215623192

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Formula

G.f.: sqrt( (1-3*x)/(1-7*x) ).

n*a(n) = 2*(5*n-4)*a(n-1) - 21*(n-2)*a(n-2).

Sum_{i=0..n} Sum_{j=0..i} (1/3)^i * a(j) * a(i-j) = (7/3)^n.

a(n) = 2 * A122898(n-1) for n > 0.

a(n) ~ 2 * 7^(n - 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 04 2023

Prog

(PARI) a(n) = sum(k=0, n, 3^(n-k)*binomial(n-1, n-k)*binomial(2*k, k));
(PARI) my(N=30, x='x+O('x^N)); Vec(sqrt((1-3*x)/(1-7*x)))

Crossrefs

Cf. A063886, A085362, A360317, A360319, A360321, A360322.

Cf. A085364, A122898.

Sequence in context: A020004 A078469 A014351 * A074616 A370242 A352373

Adjacent sequences: A360315 A360316 A360317 * A360319 A360320 A360321

Keyword

nonn,easy

Author

Seiichi Manyama, Feb 03 2023

Status

approved