A357243 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(1 - x).

1, 1, -2, 6, -52, 540, -7608, 129304, -2612608, 60867360, -1608663840, 47527158624, -1552431588288, 55547889458880, -2160724031160576, 90782738645280000, -4097139872604807168, 197675862365363088384, -10153243488783257091072

Offset

0,3

Links

Table of n, a(n) for n=0..18.

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

Formula

E.g.f. satisfies A(x)^A(x) * (1 - x)^(1 - x) = 1.

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

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

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

Mathematica

nmax = 20; A[_] = 1;
Do[A[x_] = ((1 - x)^(-1 + x))^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

Prog

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

Crossrefs

Cf. A005727, A155456, A349561, A356905, A356908.

Sequence in context: A027263 A320453 A277477 * A277363 A156340 A337510

Adjacent sequences: A357240 A357241 A357242 * A357244 A357245 A357246

Keyword

sign

Author

Seiichi Manyama, Sep 19 2022

Status

approved