A362380 E.g.f. satisfies A(x) = exp(x + 3*x^2/2 * A(x)).

1, 1, 4, 19, 154, 1456, 18136, 260002, 4430812, 85170988, 1854422236, 44693165716, 1188169271488, 34434053438968, 1082632555160248, 36666259172292016, 1331754793762045456, 51622725829298301520, 2127683533625205288400

Offset: 0

Author

Seiichi Manyama, Apr 20 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=3 of A362377.

Cf. A362397.

Programs

Mathematica

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x + 3*x^2/2*A[x]] + 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(-3*x^2/2*exp(x)))))

Formula

E.g.f.: exp(x - LambertW(-3*x^2/2 * exp(x))) = -2 * LambertW(-3*x^2/2 * exp(x))/(3*x^2).

a(n) = n! * Sum_{k=0..floor(n/2)} (3/2)^k * (k+1)^(n-k-1) / (k! * (n-2*k)!).