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 A356797 E.g.f. satisfies log(A(x)) = x * (exp(x) - 1) * A(x)^2. 4
 1, 0, 2, 3, 64, 305, 6936, 64897, 1645008, 24290289, 692240680, 14243244521, 456748635432, 12105737521033, 435619742434800, 14112089558682585, 567134312211275296, 21653262317886286817, 966207399513747354072, 42358800314758614030505 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Table of n, a(n) for n=0..19. Eric Weisstein's World of Mathematics, Lambert W-Function. FORMULA a(n) = n! * Sum_{k=0..floor(n/2)} (2*k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!. E.g.f.: A(x) = Sum_{k>=0} (2*k+1)^(k-1) * (x * (exp(x) - 1))^k / k!. E.g.f.: A(x) = exp( -LambertW(2 * x * (1 - exp(x)))/2 ). E.g.f.: A(x) = ( LambertW(2 * x * (1 - exp(x)))/(2 * x * (1 - exp(x))) )^(1/2). MATHEMATICA m = 20; (* number of terms *) CoefficientList[Exp[-(1/2)*LambertW[-2*(Exp[x]-1)*x]] + O[x]^m, x]*Range[0, m-1]! (* Jean-François Alcover, Sep 11 2022 *) PROG (PARI) a(n) = n!*sum(k=0, n\2, (2*k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!); (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (2*k+1)^(k-1)*(x*(exp(x)-1))^k/k!))) (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(2*x*(1-exp(x)))/2))) (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(2*x*(1-exp(x)))/(2*x*(1-exp(x))))^(1/2))) CROSSREFS Cf. A052506, A355843, A356798. Cf. A356788. Sequence in context: A280582 A037426 A004854 * A356785 A015169 A041953 Adjacent sequences: A356794 A356795 A356796 * A356798 A356799 A356800 KEYWORD nonn AUTHOR Seiichi Manyama, Aug 28 2022 STATUS approved

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Last modified June 8 02:04 EDT 2023. Contains 363157 sequences. (Running on oeis4.)