A376494 E.g.f. satisfies A(x) = exp(x^2 * A(x)^2 / (1 - x)).

1, 0, 2, 6, 84, 720, 12000, 178920, 3744720, 79531200, 2056652640, 56284351200, 1753673423040, 58443081016320, 2142625074670080, 83948606126985600, 3549356731374854400, 159643527455123712000, 7656564912324122995200

Offset: 0

Author

Seiichi Manyama, Sep 25 2024

Keywords

• nonn

Links

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

Crossrefs

Cf. A052868, A376495.

Cf. A052845, A376474.

Programs

PARI

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

PARI

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

Formula

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

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

a(n) ~ sqrt(16 + 2*exp(-1) - 2*exp(-1/2)*sqrt(exp(-1)+8)) * (exp(1/2)*sqrt(exp(-1)+8) - 1) * 2^(2*n-2) * n^(n-1) / ((4 + exp(-1) - exp(-1/2)*sqrt(exp(-1)+8)) * (sqrt(1 + 8*exp(1)) - 1)^n). - Vaclav Kotesovec, Aug 05 2025