A368891 a(n) = Sum_{k=0..floor(n/3)} n^k * binomial(n-2*k,k).
1, 1, 1, 4, 9, 16, 61, 183, 433, 1603, 5581, 15951, 59449, 225928, 738893, 2827321, 11387617, 41174086, 163185805, 686315474, 2680560361, 11035625413, 48086847117, 199640217719, 853587430801, 3836667616201, 16739402030989, 74206353913480
Offset: 0
Programs
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Mathematica
Table[HypergeometricPFQ[{1/3 - n/3, 2/3 - n/3, -n/3}, {1/2 - n/2, -n/2}, -27*n/4], {n, 0, 30}] (* Vaclav Kotesovec, Jan 09 2024 *) -
PARI
a(n) = sum(k=0, n\3, n^k*binomial(n-2*k, k));
Formula
a(n) = [x^n] 1/(1 - x - n*x^3).
a(n) ~ exp(n^(2/3)/3 + n^(1/3)/18) * n^(n/3) / 3 * (1 + 2/(3*n^(1/3)) + 2/(9*n^(2/3))). - Vaclav Kotesovec, Jan 09 2024