A368733 - OEIS

%I #17 May 29 2024 06:58:41

%S 1,1,4,22,148,1132,9484,85066,804556,7939738,81128800,853424464,

%T 9201391456,101327618056,1136518296892,12954283592578,149770265417692,

%U 1753615603901818,20766700361401336,248449277456597908,3000039734827403608,36532024054221028576,448294209318801516064

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

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

%F 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

%t Table[HypergeometricPFQ[{-1-n, -n, 1-n}, {2, 3}, -3], {n, 0, 30}]

%t 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 *)

%o (Python)

%o from sympy import hyperexpand

%o from sympy.functions import hyper

%o def A368733(n): return hyperexpand(hyper((-1-n,-n,1-n),(2,3),-3)) # _Chai Wah Wu_, Jan 04 2024

%Y Cf. A001181, A368708.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Jan 04 2024