OFFSET
0,3
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
PROG
(PARI) a(n) = n!*sum(k=0, n\2, (n+1)^(k-1)*binomial(n-k-1, n-2*k)/k!);
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Sep 24 2024
STATUS
approved