A351769 - OEIS

%I #18 Jun 02 2022 15:38:20

%S 1,1,17,827,79368,12623124,3002832110,998401869464,442148442609152,

%T 251578963946182968,178846127724854653704,155339277405600252114072,

%U 161863497852092601156187152,199286757107586767535516731832,286210094619439661737214469710088

%N a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling1(n, k) * k^(k+n).

%H Seiichi Manyama, <a href="/A351769/b351769.txt">Table of n, a(n) for n = 0..213</a>

%F a(n) ~ c * r^n * (1 + r*exp(1 + 1/r))^n * n^(2*n) / exp(2*n), where r = 0.937997555632908331545534056235449048849427140626270261830822459734975609... is the root of the equation r + exp(-1 - 1/r) = -LambertW(-1, -r*exp(-r)) and c = 0.9367460233410089838603007174937882495902299959682250862650092226619624... - _Vaclav Kotesovec_, Feb 18 2022

%F E.g.f.: Sum_{k>=0} (-k * log(1 - k*x))^k / k!. - _Seiichi Manyama_, Jun 02 2022

%t Table[Sum[k^(k+n) * StirlingS1[n, k] * (-1)^(n-k), {k, 0, n}], {n, 0, 20}]

%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*stirling(n, k, 1)*k^(k+n)); \\ _Michel Marcus_, Feb 19 2022

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k*log(1-k*x))^k/k!))) \\ _Seiichi Manyama_, Jun 02 2022

%Y Cf. A351180, A351181.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Feb 18 2022