A364939 E.g.f. satisfies A(x) = exp( x*A(x) / (1 - x*A(x))^2 ).
1, 1, 7, 82, 1421, 32856, 953107, 33316816, 1364109273, 64057409920, 3394727354591, 200445915043584, 13050860745456613, 928976320999078912, 71773343988758253675, 5982029183718123513856, 535011546414154955711153, 51110145581257562326401024
Offset: 0
Keywords
Programs
-
Mathematica
Table[n! * Sum[(n+1)^(k-1) * Binomial[n+k-1,n-k]/k!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 11 2023 *) -
PARI
a(n) = n!*sum(k=0, n, (n+1)^(k-1)*binomial(n+k-1, n-k)/k!);
Formula
a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(n+k-1,n-k)/k!.
a(n) ~ sqrt(((321*(3852 + 215*sqrt(321)))^(1/3) - 321^(2/3)/(3852 + 215*sqrt(321))^(1/3)) / 107) * (4 + ((83 - 3*sqrt(321))/2)^(1/3) + ((83 + 3*sqrt(321))/2)^(1/3))^n * exp(((215 - 12*sqrt(321))^(1/3) + (215 + 12*sqrt(321))^(1/3) - 1) * (n+1)/12 - n) * n^(n-1) / 3^(n + 1/2). - Vaclav Kotesovec, Nov 11 2023
E.g.f.: (1/x) * Series_Reversion( x*exp(-x/(1 - x)^2) ). - Seiichi Manyama, Sep 23 2024