A376474 E.g.f. satisfies A(x) = exp( x^2*A(x)^2 / (1 - x*A(x)) ).
1, 0, 2, 6, 84, 840, 14160, 246960, 5438160, 132209280, 3696265440, 114042297600, 3898083752640, 145315002792960, 5886559994515200, 257081021880883200, 12051082491262214400, 603307920100773888000, 32132914081702520486400, 1814085935013542141952000, 108218538908648830498636800
Offset: 0
Keywords
Programs
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PARI
a(n) = n!*sum(k=0, n\2, (n+1)^(k-1)*binomial(n-k-1, n-2*k)/k!);
Formula
E.g.f.: (1/x) * Series_Reversion( x*exp(-x^2 / (1 - x)) ).
a(n) = n! * Sum_{k=0..floor(n/2)} (n+1)^(k-1) * binomial(n-k-1,n-2*k)/k!.
a(n) ~ s^2 * (2-r*s) * n^(n-1) / (sqrt(2 - 2*r*s + 4*r^2*s^2 - 4*r^3*s^3 + r^4*s^4) * r^(n-1) * exp(n)), where r = exp(1 - sqrt(7/3) * cos(arctan(3^(-3/2))/3) + sqrt(7) * sin(arctan(3^(-3/2))/3)) * ((1 + sqrt(7) * cos(arctan(3^(3/2))/3) - sqrt(21) * sin(arctan(3^(3/2))/3))/3) = 0.311460490854501594554904428274272083649... and s = exp(-1 + sqrt(7/3) * cos(arctan(3^(-3/2))/3) - sqrt(7) * sin(arctan(3^(-3/2))/3)) = 1.428887069084244135127491236860585605773... - Vaclav Kotesovec, Sep 24 2024