A357245 E.g.f. satisfies A(x) * log(A(x)) = 3 * (exp(x) - 1).
1, 3, -6, 84, -1599, 42906, -1477716, 62171661, -3090518556, 177237143040, -11518529575857, 836601742598628, -67156626492464064, 5904119985344031639, -564188922815428792914, 58225175660113940932032, -6453955474121138652732903, 764716767229825444834522086
Offset: 0
Keywords
Links
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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Mathematica
nmax = 17; A[_] = 1; Do[A[x_] = Exp[(3*(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, 3^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)*(3*(exp(x)-1))^k/k!))) -
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(3*(exp(x)-1))))) -
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(3*(exp(x)-1)/lambertw(3*(exp(x)-1))))
Formula
a(n) = Sum_{k=0..n} 3^k * (-k+1)^(k-1) * Stirling2(n,k).
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (3 * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( LambertW(3 * (exp(x) - 1)) ).
E.g.f.: A(x) = 3 * (exp(x) - 1)/LambertW(3 * (exp(x) - 1)).