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A356912
E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^2/2).
3
1, 0, 0, 3, 6, 20, 0, -126, -1260, 18360, 335160, 4546080, 26302320, -59501520, -5703994296, -58549768200, 371346066000, 34962417322560, 746101280831040, 8059680118183680, -93772611412099200, -5613314502242643840, -110940169654432087200
OFFSET
0,4
LINKS
Eric Weisstein's World of Mathematics, Lambert W-Function.
FORMULA
a(n) = n! * Sum_{k=0..floor(n/3)} (-k+1)^(k-1) * |Stirling1(n-2*k,k)|/(2^k * (n-2*k)!).
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-x^2/2 * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-x^2/2 * log(1-x)) ).
E.g.f.: A(x) = -x^2/2 * log(1-x)/LambertW(-x^2/2 * log(1-x)).
MATHEMATICA
nmax = 22; A[_] = 1;
Do[A[x_] = ((1 - x)^(-x^2/2))^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
PROG
(PARI) a(n) = n!*sum(k=0, n\3, (-k+1)^(k-1)*abs(stirling(n-2*k, k, 1))/(2^k*(n-2*k)!));
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-x^2/2*log(1-x))^k/k!)))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x^2/2*log(1-x)))))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-x^2/2*log(1-x)/lambertw(-x^2/2*log(1-x))))
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Sep 03 2022
STATUS
approved