A360465 E.g.f. satisfies A(x) = exp(x * exp(2*x) * A(x)).

1, 1, 7, 64, 829, 14056, 295399, 7426252, 217637305, 7291538704, 275050426411, 11540336658676, 533224609095061, 26908386824872216, 1472691380336896399, 86892807951798473116, 5498668489586321670769, 371511527654280649783840

Offset: 0

Author

Seiichi Manyama, Feb 08 2023

Keywords

• nonn

Links

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

Crossrefs

Cf. A038050, A273954, A357247, A360466.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*exp(2*x)))))

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(-lambertw(-x*exp(2*x))/(x*exp(2*x))))

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x*exp(2*x))^k/k!)))

PARI

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

Formula

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

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

E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (x * exp(2*x))^k / k!.

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

a(n) ~ sqrt(1+LambertW(2*exp(-1))) * 2^n * n^(n-1) / (exp(n-1) * LambertW(2*exp(-1))^n). - Vaclav Kotesovec, Feb 08 2023