A352946 a(n) = Sum_{k=0..floor(n/3)} (n-3*k)^k.
1, 1, 1, 1, 2, 3, 4, 6, 10, 16, 25, 42, 73, 125, 217, 391, 714, 1305, 2428, 4612, 8830, 17038, 33377, 66216, 132349, 267075, 545329, 1123693, 2333278, 4889751, 10342468, 22043954, 47340802, 102504532, 223654713, 491393646, 1087353601, 2423448817, 5437568233
Offset: 0
Programs
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PARI
a(n) = sum(k=0, n\3, (n-3*k)^k);
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PARI
my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k*x^3)))
Formula
G.f.: Sum_{k>=0} x^k / (1 - k * x^3).
a(n) ~ sqrt(2*Pi/3) * (n/LambertW(exp(1)*n))^(n*(1 - 1/LambertW(exp(1)*n))/3 + 1/2) / sqrt(1 + LambertW(exp(1)*n)). - Vaclav Kotesovec, Apr 14 2022