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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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Table of n, a(n) for n=0..22. 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 Cf. A351492, A356752. Cf. A356909, A356910. Sequence in context: A306522 A290784 A355605 * A176993 A359963 A276748 Adjacent sequences: A356909 A356910 A356911 * A356913 A356914 A356915 KEYWORD sign AUTHOR Seiichi Manyama, Sep 03 2022 STATUS approved

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Last modified May 23 02:30 EDT 2024. Contains 372758 sequences. (Running on oeis4.)