A382036 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * C(x)^2) ), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.
1, 1, 7, 94, 1901, 51696, 1771267, 73317616, 3560476761, 198531343360, 12502959204671, 877829600807424, 67991178144166213, 5759309535250776064, 529665762441463234875, 52560256640090731902976, 5597859153748148214250673, 636915477940535101583130624, 77102760978489789146276986231
Offset: 0
Keywords
Programs
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PARI
a(n) = if(n==0, 1, (n-1)!*sum(k=0, n-1, (n+1)^(n-k-1)*binomial(2*n, k)/(n-k-1)!));
Formula
E.g.f. A(x) satisfies A(x) = exp(x*A(x) * C(x*A(x))^2).
a(n) = (n-1)! * Sum_{k=0..n-1} (n+1)^(n-k-1) * binomial(2*n,k)/(n-k-1)! for n > 0.
E.g.f.: exp( Series_Reversion( x*exp(-x)/(1+x)^2 ) ).
a(n) ~ 2^(n - 1/2) * n^(n-1) / ((sqrt(2) - 1)^(n - 1/2) * exp((sqrt(2) - 1)*(sqrt(2)*n - 1))). - Vaclav Kotesovec, Mar 15 2025