A358113 a(n) = 16^n * Sum_{k=0..n} (-1)^k * binomial(1/2, k)^2 * binomial(n, k).

1, 12, 132, 1200, 5220, -132048, -5451376, -139104576, -3034129500, -61171843920, -1176294856176, -21916435874112, -399241706218992, -7151078337480000, -126420386691188160, -2211675290036790528, -38363623542890191836, -660751288131343246416, -11312478475520480652400

Offset

0,2

Comments

Belongs to the family of Apéry-like sequences.

Links

Table of n, a(n) for n=0..18.

Formula

a(n) = 16^n * hypergeom([-1/2, -1/2, -n], [1, 1], 1).

From Vaclav Kotesovec, Nov 12 2022: (Start)

G.f.: LegendreP(1/2, 1 - 32*x) / (1 - 16*x)^(3/2).

Recurrence: n^2*a(n) = 4*(8*n^2 - 5)*a(n-1) - 256*(n-1)*(n+1)*a(n-2).

a(n) ~ -2^(4*n + 1) * sqrt(n) * log(n) / Pi^(3/2) * (1 - c/log(n)), where c = 1.2639012517387952828900951811685381605048398578985... (End)

c = 6 - 6*log(2) - gamma, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 18 2024

Maple

a := n -> 16^n*add((-1)^k*binomial(1/2, k)^2*binomial(n, k), k = 0..n):
seq(a(n), n = 0..18);

Mathematica

a[n_] := 16^n * HypergeometricPFQ[{-1/2, -1/2, -n}, {1, 1}, 1]; Array[a, 19, 0] (* Amiram Eldar, Nov 12 2022 *)
CoefficientList[Series[LegendreP[1/2, 1 - 32*x]/(1 - 16*x)^(3/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 12 2022 *)

Crossrefs

Cf. A358108, A358109, A143583.

Sequence in context: A002721 A340508 A119217 * A334334 A119237 A038729

Adjacent sequences: A358110 A358111 A358112 * A358114 A358115 A358116

Keyword

sign

Author

Peter Luschny, Nov 12 2022

Status

approved