A362524 a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) / (2^k * k! * (n-2*k)!).

1, 1, 2, 4, 16, 56, 391, 2017, 20504, 139456, 1867681, 15751451, 262263442, 2638794094, 52589415971, 614628436801, 14274125637256, 190012483804952, 5041005195499849, 75288391385094811, 2246914521052963166, 37204717212894726706, 1233884675800841217847

Offset: 0

Author

Seiichi Manyama, Apr 23 2023

Keywords

• nonn
• easy

Links

• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A088957, A362525.

Cf. A362350, A362522.

Programs

Mathematica

Table[n!Sum[(k+1)^(k-1)/(2^k k!(n-2k)!),{k,0,Floor[n/2]}],{n,0,25}] (* Harvey P. Dale, Mar 30 2024 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^2/2))))

Formula

E.g.f.: exp(x - LambertW(-x^2/2)) = -2 * LambertW(-x^2/2)/x^2 * exp(x).

a(n) ~ (exp(sqrt(2)*exp(-1/2) + 1) + (-1)^n*exp(1 - sqrt(2)*exp(-1/2))) * n^(n-1) / (2^((n-1)/2) * exp(n/2)). - Vaclav Kotesovec, Aug 05 2025