A357026 E.g.f. satisfies A(x) = (1 - x)^(log(1 - x) * A(x)).
1, 0, 2, 6, 58, 460, 5528, 70308, 1098060, 18910512, 371480832, 8022952080, 191325228576, 4961955705408, 139572074260656, 4224646630879920, 137077496211066384, 4744151145076980864, 174517898073769832448, 6798949897214608689024, 279688643858492900930496
Offset: 0
Keywords
Links
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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Mathematica
m = 21; (* number of terms *) A[_] = 0; Do[A[x_] = (1 - x)^(Log[1 - x]*A[x]) + O[x]^m // Normal, {m}]; CoefficientList[A[x], x]*Range[0, m - 1]! (* Jean-François Alcover, Sep 12 2022 *) -
PARI
a(n) = sum(k=0, n\2, (2*k)!*(k+1)^(k-1)*abs(stirling(n, 2*k, 1))/k!);
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PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*log(1-x)^(2*k)/k!))) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-log(1-x)^2)))) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(-lambertw(-log(1-x)^2)/log(1-x)^2))
Formula
E.g.f. satisfies log(A(x)) = log(1 - x)^2 * A(x).
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * (k+1)^(k-1) * |Stirling1(n,2*k)|/k!.
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * log(1 - x)^(2*k) / k!.
E.g.f.: A(x) = exp( -LambertW(-log(1-x)^2) ).
E.g.f.: A(x) = -LambertW(-log(1 - x)^2)/log(1 - x)^2.
a(n) ~ sqrt(2) * exp(5/4 + exp(-1/2)/2 - n) * n^(n-1) / (1 - exp(-exp(-1/2)))^(n - 1/2). - Vaclav Kotesovec, Aug 05 2025