A349653 E.g.f. satisfies: A(x)^(A(x)^3) = 1/(1 - x).

1, 1, -4, 51, -996, 27120, -943602, 40023354, -2002953432, 115536775248, -7547711366880, 550798542893808, -44409102801760584, 3920444594317227600, -376109365694009875704, 38961901445878423746360, -4334496557343337848950208, 515407133679990302374396416

Offset: 0

Author

Seiichi Manyama, Nov 23 2021

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..336
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A349651, A349655, A349657.

Cf. A349561, A349652.

Programs

Mathematica

nmax = 20; A[_] = 1;
Do[A[x_] = (1/(1 - x))^(1/A[x]^3) + 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, (3*k-1)^(k-1)*stirling(n, k, 1));

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=0, N, (3*k-1)^(k-1)*log(1-x)^k/k!)))

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace((-3*log(1-x)/lambertw(-3*log(1-x)))^(1/3)))

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} (3*k-1)^(k-1) * Stirling1(n,k).

E.g.f.: A(x) = -Sum_{k>=0} (3*k-1)^(k-1) * (log(1-x))^k / k!.

E.g.f.: A(x) = ( -3*log(1-x)/LambertW(-3*log(1-x)) )^(1/3).

a(n) ~ -(-1)^n * exp(1/6 - exp(-1)/6 - n) * n^(n-1) / (sqrt(3) * (-1 + exp(exp(-1)/3))^(n - 1/2)). - Vaclav Kotesovec, Nov 24 2021