A360317 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

1, 2, 10, 52, 278, 1516, 8388, 46920, 264678, 1503052, 8581676, 49215256, 283297660, 1635904376, 9472214344, 54975423504, 319729353606, 1862896455180, 10871759717916, 63539265366264, 371837338366740, 2178604586281128, 12778264475444280, 75022726995053808

Offset

0,2

Links

Table of n, a(n) for n=0..23.

Formula

G.f.: sqrt( (1-2*x)/(1-6*x) ).

n*a(n) = 2*(4*n-3)*a(n-1) - 12*(n-2)*a(n-2).

Sum_{i=0..n} Sum_{j=0..i} (1/2)^i * a(j) * a(i-j) = 3^n.

a(n) = 2 * A005573(n-1) for n > 0.

a(n) ~ 2^(n + 1/2) * 3^(n - 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 04 2023

Prog

(PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(n-1, n-k)*binomial(2*k, k));
(PARI) my(N=30, x='x+O('x^N)); Vec(sqrt((1-2*x)/(1-6*x)))

Crossrefs

Cf. A063886, A085362, A360318, A360319, A360321, A360322.

Cf. A005573, A085363.

Sequence in context: A319325 A361805 A344576 * A074612 A104497 A166694

Adjacent sequences: A360314 A360315 A360316 * A360318 A360319 A360320

Keyword

nonn,easy

Author

Seiichi Manyama, Feb 03 2023

Status

approved