A380406 E.g.f. satisfies A(x) = exp( 2 * x * exp(x) * A(x)^(1/2) ).

1, 2, 12, 104, 1232, 18592, 342208, 7451264, 187631872, 5369721344, 172255038464, 6125052946432, 239195824279552, 10179739052908544, 469024768235192320, 23263095316577681408, 1235978286454556131328, 70040404736026578386944, 4217180561907991530176512

Offset: 0

Author

Seiichi Manyama, Jan 23 2025

Keywords

• nonn

Links

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

Crossrefs

Cf. A273954, A357247, A380407.

Cf. A273953, A360473.

Programs

PARI

a(n) = 2*sum(k=0, n, k^(n-k)*(k+2)^(k-1)*binomial(n, k));

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A273954.

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

a(n) = 2 * Sum_{k=0..n} k^(n-k) * (k+2)^(k-1) * binomial(n,k).

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