A356910 - OEIS

%I #16 Feb 16 2025 08:34:03

%S 1,0,0,6,12,40,-180,-1512,-11760,142560,2701440,37033920,-47472480,

%T -7299227520,-181704466944,-904179830400,40024286265600,

%U 1774386897454080,24426730612869120,-217650777809310720,-26326923875473536000,-662608157128469637120

%N E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^2).

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

%F a(n) = n! * Sum_{k=0..floor(n/3)} (-k+1)^(k-1) * |Stirling1(n-2*k,k)|/(n-2*k)!.

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

%F E.g.f.: A(x) = exp( LambertW(-x^2 * log(1-x)) ).

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

%t nmax = 21; A[_] = 1;

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

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

%o (PARI) a(n) = n!*sum(k=0, n\3, (-k+1)^(k-1)*abs(stirling(n-2*k, k, 1))/(n-2*k)!);

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-x^2*log(1-x))^k/k!)))

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x^2*log(1-x)))))

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-x^2*log(1-x)/lambertw(-x^2*log(1-x))))

%Y Cf. A356905, A356911.

%Y Cf. A353228, A355287.

%K sign,changed

%O 0,4

%A _Seiichi Manyama_, Sep 03 2022