A349653 - OEIS

%I #25 Mar 04 2024 04:40:14

%S 1,1,-4,51,-996,27120,-943602,40023354,-2002953432,115536775248,

%T -7547711366880,550798542893808,-44409102801760584,

%U 3920444594317227600,-376109365694009875704,38961901445878423746360,-4334496557343337848950208,515407133679990302374396416

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

%H Seiichi Manyama, <a href="/A349653/b349653.txt">Table of n, a(n) for n = 0..336</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

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

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

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

%F 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

%t nmax = 20; A[_] = 1;

%t Do[A[x_] = (1/(1 - x))^(1/A[x]^3) + O[x]^(nmax+1) // Normal, {nmax}];

%t CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)

%o (PARI) a(n) = (-1)^(n-1)*sum(k=0, n, (3*k-1)^(k-1)*stirling(n, k, 1));

%o (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!)))

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

%Y Cf. A349651, A349655, A349657.

%Y Cf. A349561, A349652.

%K sign

%O 0,3

%A _Seiichi Manyama_, Nov 23 2021