login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

E.g.f. satisfies A(x) = exp(x + x^3/3 * A(x)^3).
6

%I #20 Mar 04 2024 08:47:27

%S 1,1,1,3,33,321,2841,31641,498849,8979489,167510961,3427780401,

%T 80374833441,2089382321313,58020408889353,1721768971537161,

%U 55150870311938241,1897482353016075201,69322763655015214689,2676706914491568918369

%N E.g.f. satisfies A(x) = exp(x + x^3/3 * A(x)^3).

%H Seiichi Manyama, <a href="/A362478/b362478.txt">Table of n, a(n) for n = 0..399</a>

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

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

%F a(n) = n! * Sum_{k=0..floor(n/3)} (1/3)^k * (3*k+1)^(n-2*k-1) / (k! * (n-3*k)!).

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

%t Do[A[x_] = Exp[x + x^3/3*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) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^3*exp(3*x))/3)))

%Y Column k=2 of A362490.

%Y Cf. A362474, A362491.

%Y Cf. A362390.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Apr 21 2023