A357246 E.g.f. satisfies A(x) * log(A(x)) = (1-x) * (exp(x) - 1).

1, 1, -2, 5, -49, 497, -6926, 116510, -2325422, 53538315, -1397740279, 40792008435, -1316056239994, 46509292766172, -1786748828967402, 74139054468535061, -3304409577659864305, 157444695280699565069, -7986085592316390890618, 429645521271113815480246

Offset: 0

Author

Seiichi Manyama, Sep 19 2022

Keywords

• sign

Links

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

Crossrefs

Cf. A356902, A357243, A357247.

Programs

Mathematica

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

PARI

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

PARI

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

PARI

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

Formula

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

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

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