A364941 E.g.f. satisfies A(x) = exp( x*A(x)^2 / (1 - x*A(x))^2 ).
1, 1, 9, 139, 3201, 98861, 3842653, 180342471, 9926870145, 627296384665, 44766115252821, 3561306199330859, 312531347680052449, 29994317717748851013, 3125271184480991706189, 351360521075659460743471, 42395667639523579933634817, 5464885215245368415146646321
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Keywords
Programs
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Mathematica
Join[{1}, Table[n! * Sum[(n+k+1)^(k-1) * Binomial[n+k-1,n-k]/k!, {k,0,n}], {n,1,20}]] (* Vaclav Kotesovec, Nov 18 2023 *) -
PARI
a(n) = n!*sum(k=0, n, (n+k+1)^(k-1)*binomial(n+k-1, n-k)/k!);
Formula
a(n) = n! * Sum_{k=0..n} (n+k+1)^(k-1) * binomial(n+k-1,n-k)/k!.
a(n) ~ s^2 * sqrt((1 + r*s)/(1 + 2*r*s^2 - 3*r^2*s^2 + 2*r^3*s^3)) * n^(n-1) / (exp(n) * r^(n - 1/2)), where r = 0.1208150626316801846776206051780724146363... and s = 1.505405324736640697527292770220289316454393380356... are real roots of the system of equations exp(r*s^2 / (1 - r*s)^2) = s, 2*r*s^2 = (1 - r*s)^3. - Vaclav Kotesovec, Nov 18 2023