A352082 a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^n.
1, 1, 4, 28, 272, 3369, 50816, 903856, 18522624, 429746905, 11135257600, 318719062236, 9987013488640, 340037795872369, 12500401969233920, 493467700789897408, 20819865970795610112, 934939160745193002321, 44523294861684890664960
Offset: 0
Programs
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Mathematica
a[0] = 1; a[n_] := Sum[(n-2*k)^n, {k, 0, Floor[n/2]}]; Array[a, 20, 0] (* Amiram Eldar, Apr 16 2022 *) -
PARI
a(n) = sum(k=0, n\2, (n-2*k)^n);
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PARI
my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-(k*x)^2)))
Formula
G.f.: Sum_{k>=0} (k * x)^k / (1 - (k * x)^2).
Conjecture: a(n) = (1 - 2^n)*zeta(-n) - (2^n)*zeta(-n, n/2 + 1) for n > 0, where the bivariate zeta function is the Hurwitz zeta function. - Velin Yanev, Mar 25 2024
a(n) ~ n^n / (1 - exp(-2)). - Vaclav Kotesovec, Mar 25 2024