A356908 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^2.

1, 2, -2, 24, -272, 4840, -107496, 2934400, -94501760, 3511914624, -147882696960, 6959217277056, -361941871760256, 20616017227643136, -1276341094954066176, 85337532623368181760, -6128269375791673718784, 470426144496265208979456

Offset: 0

Author

Seiichi Manyama, Sep 03 2022

Keywords

• sign

Links

• Robert Israel, Table of n, a(n) for n = 0..350
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A349561, A355779, A356909.

Programs

Maple

S:= series(exp( LambertW(-2 * log(1-x))),x,51):
seq(n!*coeff(S,x,n),n=0..50); # Robert Israel, Sep 18 2022

Mathematica

nmax = 20; A[_] = 1;
Do[A[x_] = (1/(1-x)^2)^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)

PARI

a(n) = sum(k=0, n, 2^k*(-k+1)^(k-1)*abs(stirling(n, k, 1)));

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-2*log(1-x))^k/k!)))

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-2*log(1-x)))))

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(-2*log(1-x)/lambertw(-2*log(1-x))))

Formula

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

E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-2 * log(1-x))^k / k!.

E.g.f.: A(x) = exp( LambertW(-2 * log(1-x)) ).

E.g.f.: A(x) = -2 * log(1-x)/LambertW(-2 * log(1-x)).