A356798 E.g.f. satisfies log(A(x)) = x * (exp(x) - 1) * A(x)^3.

1, 0, 2, 3, 88, 425, 13476, 130417, 4543120, 71005041, 2723297860, 60685651961, 2564091428856, 75166650583609, 3496499475113932, 127585829832674865, 6521845096842043936, 284745004488498858209, 15950013722559213419412, 809403234909367349670409

Offset: 0

Author

Seiichi Manyama, Aug 28 2022

Keywords

• nonn

Links

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

Crossrefs

Cf. A052506, A355843, A356797.

Cf. A356789.

Programs

Mathematica

nmax = 19; A[_] = 1;
Do[A[x_] = Exp[x*(Exp[x] - 1)*A[x]^3] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)

PARI

a(n) = n!*sum(k=0, n\2, (3*k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (3*k+1)^(k-1)*(x*(exp(x)-1))^k/k!)))

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(3*x*(1-exp(x)))/3)))

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(3*x*(1-exp(x)))/(3*x*(1-exp(x))))^(1/3)))

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (3*k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!.

E.g.f.: A(x) = Sum_{k>=0} (3*k+1)^(k-1) * (x * (exp(x) - 1))^k / k!.

E.g.f.: A(x) = exp( -LambertW(3 * x * (1 - exp(x)))/3 ).

E.g.f.: A(x) = ( LambertW(3 * x * (1 - exp(x)))/(3 * x * (1 - exp(x))) )^(1/3).