A356905 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^x.

1, 0, 2, 3, -4, -30, 294, 3780, -7904, -444528, 78840, 99657360, 539299848, -27852945120, -361237078944, 10124338180320, 258341121976320, -4020500134465920, -205187357182405824, 1330097523844832640, 186823640933648588160, 500469438126120583680

Offset

0,3

Links

Table of n, a(n) for n=0..21.

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (-k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.

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

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

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

Mathematica

nmax = 21; A[_] = 1;
Do[A[x_] = (1/(1 - x)^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) a(n) = n!*sum(k=0, n\2, (-k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!);
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-x*log(1-x))^k/k!)))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x*log(1-x)))))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-x*log(1-x)/lambertw(-x*log(1-x))))

Crossrefs

Cf. A355842, A356795, A356796, A356906.

Cf. A349561, A356884.

Sequence in context: A191471 A354729 A305205 * A064889 A351999 A356885

Adjacent sequences: A356902 A356903 A356904 * A356906 A356907 A356908

Keyword

sign

Author

Seiichi Manyama, Sep 03 2022

Status

approved