A368898 - OEIS

%I #13 Jan 09 2024 08:50:09

%S 1,1,1,1,5,11,19,29,105,298,671,1299,3997,12468,33083,75781,220625,

%T 708867,2086183,5412778,15756741,51093316,160523859,457283931,

%U 1365001273,4458076176,14608351135,44649287452,137979763181,455582050840,1536403659211,4953147876189

%N a(n) = Sum_{k=0..floor(n/4)} n^k * binomial(n-3*k,k).

%F a(n) = [x^n] 1/(1 - x - n*x^4).

%F a(n) = hypergeom([(1-n)/4, (2-n)/4, (3-n)/4, -n/4], [(1-n)/3, (2-n)/3, -n/3], -256*n/27). - _Stefano Spezia_, Jan 09 2024

%F a(n) ~ (1/4) * exp(n^(3/4)/4 + sqrt(n)/16 + 5*n^(1/4)/384) * n^(n/4) * (1 + 30643/(40960*n^(1/4)) + 3749229947/(10066329600*sqrt(n)) + 15892274778169/(137438953472000*n^(3/4))). - _Vaclav Kotesovec_, Jan 09 2024

%t Table[HypergeometricPFQ[{1/4 - n/4, 1/2 - n/4, 3/4 - n/4, -n/4}, {1/3 - n/3, 2/3 - n/3, -n/3}, -256*n/27], {n, 0, 20}] (* _Vaclav Kotesovec_, Jan 09 2024 *)

%o (PARI) a(n) = sum(k=0, n\4, n^k*binomial(n-3*k, k));

%Y Cf. A171180, A368891.

%K nonn,easy

%O 0,5

%A _Seiichi Manyama_, Jan 09 2024