A360592 G.f.: Sum_{k>=0} (1 + k*x)^k * x^k.

1, 1, 2, 5, 14, 44, 149, 543, 2096, 8539, 36444, 162380, 752181, 3612037, 17933038, 91843329, 484280386, 2624400428, 14595111277, 83178971707, 485218783724, 2893881790823, 17628815344600, 109585578277012, 694575012732989, 4485139961090153, 29486515600393930

Offset: 0

Author

Vaclav Kotesovec, Feb 13 2023

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..760

Crossrefs

Cf. A360232, A092366, A187018, A360699, A360707.

Programs

Maple

N:= 40:
S:= series(add((1+k*x)^k*x^k, k=0..N),x,N+1):
seq(coeff(S,x,k),k=0..N); # Robert Israel, Feb 13 2023

Mathematica

nmax = 30; CoefficientList[Series[Sum[(1 + k*x)^k * x^k, {k, 0, nmax}], {x, 0, nmax}], x]
Flatten[{1, Table[Sum[Binomial[n-k, k] * (n-k)^k, {k, 0, n/2}], {n, 1, 30}]}]

PARI

{a(n) = polcoeff(sum(m=0, n, (1 + m*x)^m * x^m + x*O(x^n)), n)};
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k) * (n-k)^k.

a(n) ~ exp(exp(1/2)*sqrt(n/2) - 3*exp(1)/8) * n^(n/2) / 2^(n/2 + 1) * (1 + ((exp(1/2) + exp(-1/2))/2^(5/2) + 11*exp(3/2)/2^(9/2))/sqrt(n)).