A352944 a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^k.
1, 1, 1, 2, 3, 5, 9, 16, 31, 61, 125, 266, 579, 1305, 3009, 7120, 17255, 42697, 108005, 278466, 731883, 1958589, 5331625, 14758720, 41501135, 118507301, 343405709, 1009313322, 3007557523, 9081204849, 27775308049, 86014412384, 269603741111, 855012176081
Offset: 0
Programs
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Mathematica
Join[{1},Table[Sum[(n-2k)^k,{k,0,Floor[n/2]}],{n,40}]] (* Harvey P. Dale, Dec 12 2022 *) -
PARI
a(n) = sum(k=0, n\2, (n-2*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^2)))
Formula
G.f.: Sum_{k>=0} x^k / (1 - k * x^2).
a(n) ~ sqrt(Pi) * (n/LambertW(exp(1)*n))^((n + 1 - n/LambertW(exp(1)*n))/2) / sqrt(1 + LambertW(exp(1)*n)). - Vaclav Kotesovec, Apr 14 2022