A355843 E.g.f. satisfies log(A(x)) = x * (exp(x) - 1) * A(x).
1, 0, 2, 3, 40, 185, 2556, 22057, 349616, 4519377, 83642860, 1439639201, 31015493928, 663158322697, 16468280168900, 418772642545545, 11847925722273376, 348085509493265825, 11091199095506163420, 368912674236287743633, 13099432280183074041560
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Keywords
Programs
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Mathematica
nmax = 20; A[_] = 1; Do[A[x_] = Exp[x*(Exp[x] - 1)*A[x]] + O[x]^(nmax+1) // Normal, {nmax}]; CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *) -
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x*(1-exp(x)))))) -
PARI
a(n) = n!*sum(k=0, n\2, (k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!); \\ Seiichi Manyama, Aug 28 2022
Formula
E.g.f.: exp( -LambertW(x * (1 - exp(x))) ).
E.g.f.: LambertW(x * (1 - exp(x))) / (x * (1 - exp(x))).
a(n) ~ sqrt(1 + exp(1+r)*r^2) * n^(n-1) / (exp(n-1) * r^n), where r = 0.528399250336668412340528181936966763473482889289226687323... is the root of the equation exp(1+r) - exp(1) = 1/r. - Vaclav Kotesovec, Jul 21 2022
a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!. - Seiichi Manyama, Aug 28 2022