A374487 Expansion of 1/(1 - 2*x - 7*x^2)^(3/2).

1, 3, 18, 70, 315, 1281, 5348, 21708, 88245, 355135, 1425270, 5692050, 22666735, 89986365, 356400840, 1408459928, 5555679849, 21877337979, 86020384730, 337769595870, 1324677499299, 5189411915897, 20308936981932, 79406140870500, 310206869770525, 1210898719869111

Offset

0,2

Links

Paolo Xausa, Table of n, a(n) for n = 0..1000

Formula

a(0) = 1, a(1) = 3; a(n) = ((2*n+1)*a(n-1) + 7*(n+1)*a(n-2))/n.

a(n) = binomial(n+2,2) * A025235(n).

From Seiichi Manyama, Aug 20 2025: (Start)

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

a(n) = Sum_{k=0..n} (1/2)^k * (7/2)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k). (End)

a(n) ~ sqrt(n) * (1 + 2*sqrt(2))^(n + 3/2) / (2^(11/4) * sqrt(Pi)). - Vaclav Kotesovec, Aug 21 2025

Mathematica

Module[{x}, CoefficientList[Series[1/(1 - (7*x + 2)*x)^(3/2), {x, 0, 25}], x]] (* Paolo Xausa, Aug 25 2025 *)

Prog

(PARI) a(n) = binomial(n+2, 2)*sum(k=0, n\2, 2^k*binomial(n, 2*k)*binomial(2*k, k)/(k+1));

Crossrefs

Cf. A102839, A102840, A374488.

Cf. A014431, A025235, A084601.

Sequence in context: A373651 A157535 A373065 * A098522 A174764 A114633

Adjacent sequences: A374484 A374485 A374486 * A374488 A374489 A374490

Keyword

nonn

Author

Seiichi Manyama, Jul 09 2024

Status

approved