A378047 E.g.f. satisfies A(x) = (1+x)^2 * exp(x * A(x) / (1+x)).

1, 3, 11, 70, 685, 8966, 147769, 2938608, 68509193, 1832813866, 55354862701, 1863179311676, 69163696768093, 2807246931378462, 123685264726805825, 5879149578658117096, 299892675674572370065, 16340561709320173229906, 947234845622653951286485

Offset: 0

Author

Seiichi Manyama, Nov 15 2024

Keywords

• nonn

Links

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

Crossrefs

Cf. A362771, A377826.

Programs

PARI

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

Formula

E.g.f.: (1+x)^2 * exp( -LambertW(-x*(1+x)) ).

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

a(n) ~ (1 + sqrt(1 + 4*exp(-1)))^2 * sqrt(2 + 8*exp(-1) - 2*sqrt(1 + 4*exp(-1))) * 2^(n-3) * n^(n-1) / ((sqrt(1 + 4*exp(-1)) - 1)^n * exp(n - 3/2)). - Vaclav Kotesovec, Nov 15 2024