A357337 - OEIS

%I #23 Sep 10 2024 04:11:58

%S 0,1,3,26,334,5964,135228,3729872,121172560,4532603904,191869653120,

%T 9067948437888,473297792213376,27039987154142208,1678363256057198592,

%U 112467293224249912320,8092242817059530284032,622253112192770288799744

%N E.g.f. satisfies A(x) = log(1 + x) * exp(2 * A(x)).

%H Seiichi Manyama, <a href="/A357337/b357337.txt">Table of n, a(n) for n = 0..356</a>

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

%H <a href="/index/Res#revert">Index entries for reversions of series</a>

%F E.g.f.: -LambertW(-2 * log(1 + x))/2.

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

%F a(n) ~ n^(n-1) / (sqrt(2) * (exp(exp(-1)/2) - 1)^(n - 1/2) * exp(n + exp(-1)/4 - 1/2)). - _Vaclav Kotesovec_, Oct 04 2022

%F E.g.f.: Series_Reversion( exp(x * exp(-2*x)) - 1 ). - _Seiichi Manyama_, Sep 10 2024

%t nmax = 20; A[_] = 0;

%t Do[A[x_] = Log[1 + x]*Exp[2*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];

%t CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 05 2024 *)

%o (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-2*log(1+x))/2)))

%o (PARI) a(n) = sum(k=1, n, (2*k)^(k-1)*stirling(n, k, 1));

%Y Cf. A277489, A357338.

%Y Cf. A349504.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Sep 24 2022