A364938 E.g.f. satisfies A(x) = exp( x / (1 - x*A(x))^3 ).
1, 1, 7, 73, 1141, 23821, 623341, 19650793, 725478601, 30714824377, 1467394945561, 78103975313101, 4583805610661245, 294093243091237669, 20479664124384110101, 1538423857251845781841, 124007828871708989798161, 10676865465119963987425009
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Keywords
Programs
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Mathematica
Join[{1}, Table[n! * Sum[(n-k+1)^(k-1) * Binomial[n+2*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+2*k-1, n-k)/k!);
Formula
a(n) = n! * Sum_{k=0..n} (n-k+1)^(k-1) * binomial(n+2*k-1,n-k)/k!.
a(n) ~ sqrt(s*(1 + 2*r*s) / (4 + 3*r - 12*r*s + 12*r^2*s^2 - 4*r^3*s^3)) * n^(n-1) / (exp(n) * r^n), where r = 0.1811100305436879929789759231994897963241226689... and s = 1.893740207738561813713992833266450862854198944672... are real roots of the system of equations exp(r/(1 - r*s)^3) = s, 3*s*r^2 = (1 - r*s)^4. - Vaclav Kotesovec, Nov 18 2023