A372236 E.g.f. A(x) satisfies A(x) = exp( x * (1 + A(x)^(1/2)) ).

1, 2, 6, 26, 152, 1132, 10300, 111064, 1387104, 19713104, 314350064, 5560881328, 108110428288, 2291750937088, 52618613073408, 1301031907140608, 34470409922547200, 974354631630161152, 29270099764874881792, 931275451933870415104, 31285710787985504633856

Offset: 0

Author

Seiichi Manyama, Apr 23 2024

Keywords

• nonn

Links

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

Crossrefs

Cf. A007889, A371524.

Programs

PARI

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

PARI

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

PARI

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (k/2+1)^(k-1)*x^k/(1-(k/2+1)*x)^(k+1)))

Formula

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

If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) * (1 + A(x)^(u/r)) ), then a(n) = r * Sum_{k=0..n} (t*n+u*k+r)^(n-1) * binomial(n,k).

G.f.: Sum_{k>=0} (k/2+1)^(k-1) * x^k/(1 - (k/2+1)*x)^(k+1).

a(n) ~ sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (2^(n-1) * exp(n) * LambertW(exp(-1))^(n+2)). - Vaclav Kotesovec, Apr 24 2024