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

From Seiichi Manyama, Aug 22 2025: (Start)

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

a(n) = Sum_{k=0..n} (-1)^k * 6^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)

Mathematica

CoefficientList[Series[Sqrt[(1-2x)/(1-6x)], {x, 0, 30}], x] (* Harvey P. Dale, Oct 18 2025 *)

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 A379435 A104497

Adjacent sequences: A360314 A360315 A360316 * A360318 A360319 A360320

Keyword

nonn,easy

Author

Seiichi Manyama, Feb 03 2023

Status

approved