A368733 a(n) = hypergeom([-1 - n, -n, 1 - n], [2, 3], -3).

1, 1, 4, 22, 148, 1132, 9484, 85066, 804556, 7939738, 81128800, 853424464, 9201391456, 101327618056, 1136518296892, 12954283592578, 149770265417692, 1753615603901818, 20766700361401336, 248449277456597908, 3000039734827403608, 36532024054221028576, 448294209318801516064

Offset: 0

Author

Vaclav Kotesovec, Jan 04 2024

Keywords

• nonn

Crossrefs

Cf. A001181, A368708.

Programs

Mathematica

Table[HypergeometricPFQ[{-1-n, -n, 1-n}, {2, 3}, -3], {n, 0, 30}]
a[0] := 1; a[n_] := 3^n*Sum[(1/3)^k*Binomial[n + 1, k - 1]*Binomial[n + 1, k]*Binomial[n + 1, k + 1]/(Binomial[n + 1, 1]*Binomial[n + 1, 2]), {k, 1, n}]; Table[a[n], {n, 0, 22}] (* Detlef Meya, May 28 2024 *)

Python

from sympy import hyperexpand
from sympy.functions import hyper
def A368733(n): return hyperexpand(hyper((-1-n,-n,1-n),(2,3),-3)) # Chai Wah Wu, Jan 04 2024

Formula

a(n) ~ (4 + 3^(4/3) + 3^(5/3))^(n + 5/3) / (3^(11/6) * Pi * n^4).

a(0) = 1, a(n) = 3^n*Sum_{k=1..n} (1/3)^k*binomial(n + 1, k - 1)*binomial(n + 1, k)*binomial(n + 1, k + 1)/(binomial(n + 1, 1)*binomial(n + 1, 2)). - Detlef Meya, May 28 2024