A371142 E.g.f. satisfies A(x) = 1 + x*A(x)^2 * (exp(x) - 1).
1, 0, 2, 3, 52, 245, 4446, 40327, 823432, 11539593, 269030530, 5151458411, 137198342700, 3328477618765, 100753779655270, 2948537330575215, 100684293424304656, 3437897124964763921, 131334298073705194506, 5110990692709364143219, 216634434142329948020020
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Keywords
Programs
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Mathematica
nmax = 20; CoefficientList[Series[(1 - Sqrt[1 + 4*x - 4*E^x*x]) / (2*(-x + E^x*x)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 12 2024 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(2/(1+sqrt(1-4*x*(exp(x)-1))))) -
PARI
a(n) = n!*sum(k=0, n\2, (2*k)!/(k+1)!*stirling(n-k, k, 2)/(n-k)!);
Formula
E.g.f.: 2/(1 + sqrt(1-4*x*(exp(x)-1))).
a(n) = n! * Sum_{k=0..floor(n/2)} (2*k)!/(k+1)! * Stirling2(n-k,k)/(n-k)!.
a(n) ~ sqrt(2 + 2*r + 8*r^2) * n^(n-1) / (exp(n) * r^n), where r = 0.4454633431717774246390371283449007888959504623839... is the positive real root of the equation 4*r*(exp(r) - 1) = 1. - Vaclav Kotesovec, Mar 12 2024