A362474 E.g.f. satisfies A(x) = exp(x + x^2/2 * A(x)^2).

1, 1, 2, 10, 70, 646, 7576, 106744, 1761628, 33361948, 712950616, 16976294776, 445751093800, 12795850109992, 398697898011232, 13401365473319776, 483376669737381136, 18623161719254837008, 763300232417720682784, 33163224556779213475744

Offset: 0

Author

Seiichi Manyama, Apr 21 2023

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..392
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Column k=1 of A362483.

Cf. A362478, A362491.

Cf. A143768, A362475.

Programs

Mathematica

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x + x^2/2*A[x]^2] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

PARI

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

Formula

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

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

a(n) ~ sqrt(1 + LambertW(exp(-1/2))) * n^(n-1) / (sqrt(2) * exp(n) * LambertW(exp(-1/2))^(n+1)). - Vaclav Kotesovec, Nov 10 2023