A357244 E.g.f. satisfies A(x) * log(A(x)) = 2 * (exp(x) - 1).
1, 2, -2, 22, -266, 4614, -102442, 2777030, -88914730, 3283693254, -137408080298, 6425417730758, -332055079469610, 18792899306652358, -1156017201432075946, 76796076655220486854, -5479395288838822143786, 417905042599836811225798, -33928512587303405767179178
Offset: 0
Keywords
Links
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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Mathematica
nmax = 18; A[_] = 1; Do[A[x_] = Exp[(2*(Exp[x] - 1))/A[x]] + O[x]^(nmax+1) // Normal, {nmax}]; CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *) -
PARI
a(n) = sum(k=0, n, 2^k*(-k+1)^(k-1)*stirling(n, k, 2));
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PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(2*(exp(x)-1))^k/k!))) -
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(2*(exp(x)-1))))) -
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(2*(exp(x)-1)/lambertw(2*(exp(x)-1))))
Formula
a(n) = Sum_{k=0..n} 2^k * (-k+1)^(k-1) * Stirling2(n,k).
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (2 * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( LambertW(2 * (exp(x) - 1)) ).
E.g.f.: A(x) = 2 * (exp(x) - 1)/LambertW(2 * (exp(x) - 1)).