A349650 E.g.f. satisfies: A(x)^(A(x)^2) = 1 + x.

1, 1, -4, 36, -532, 11040, -295188, 9655772, -373422320, 16666348464, -843095987520, 47669276120928, -2979044176833888, 203906085094788960, -15170476121142482112, 1218972837861962011200, -105202043767190506428672, 9705514148732971389369600

Offset: 0

Author

Seiichi Manyama, Nov 23 2021

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..347
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A349652, A349654, A349656.

Cf. A120980, A349651.

Programs

Mathematica

nmax = 20; A[_] = 1;
Do[A[x_] = (1 + x)^(1/A[x]^2) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

PARI

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

PARI

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

PARI

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

Formula

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

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

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

a(n) ~ -(-1)^n * n^(n-1) * exp(n*(exp(-1)/2 - 1)) / (sqrt(2) * (exp(exp(-1)/2) - 1)^(n - 1/2)). - Vaclav Kotesovec, Nov 24 2021