A380407 E.g.f. satisfies A(x) = exp( 3 * x * exp(x) * A(x)^(1/3) ).

1, 3, 21, 207, 2697, 43803, 854685, 19512615, 510977937, 15112457523, 498560461989, 18160560320895, 724240913035545, 31394996915447883, 1470245245400432685, 73987438021589516247, 3982389565847576723745, 228331703268783136636515, 13894569264190369648271157

Offset: 0

Author

Seiichi Manyama, Jan 23 2025

Keywords

• nonn

Links

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

Crossrefs

Cf. A273954, A357247, A380406.

Programs

Mathematica

terms = 19; A[] = 0; Do[A[x] = Exp[3*x*Exp[x]*A[x]^(1/3)] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Jun 14 2025 *)

PARI

a(n) = 3*sum(k=0, n, k^(n-k)*(k+3)^(k-1)*binomial(n, k));

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A273954.

E.g.f.: A(x) = exp( -3*LambertW(-x * exp(x)) ).

a(n) = 3 * Sum_{k=0..n} k^(n-k) * (k+3)^(k-1) * binomial(n,k).