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A365282
E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x^2*A(x)).
4
1, 1, 2, 12, 96, 900, 10800, 157080, 2634240, 50455440, 1089849600, 26157479040, 690848040960, 19924295751360, 623024501299200, 20996216063222400, 758724126031872000, 29267547577396128000, 1200407895406514995200
OFFSET
0,3
FORMULA
a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k)^k * binomial(n-k+1,n-2*k)/( (n-k+1)*k! ).
a(n) ~ sqrt((s+1)/(2*s-1)) * (s-1)^((n+1)/2) * s^(n/2 + 1) * n^(n-1) / exp(n), where s = 3.011547791499065828694160466323712196300874261862... is the root of the equation (s-1)*LambertW(2*(s-1)^2/s) = 2. - Vaclav Kotesovec, Aug 31 2023
MATHEMATICA
Join[{1}, Table[n! * Sum[(n-2*k)^k * Binomial[n-k+1, n-2*k] / ((n-k+1)*k!), {k, 0, Floor[n/2]}], {n, 1, 20}]] (* Vaclav Kotesovec, Aug 31 2023 *)
PROG
(PARI) a(n) = n!*sum(k=0, n\2, (n-2*k)^k*binomial(n-k+1, n-2*k)/((n-k+1)*k!));
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Aug 31 2023
STATUS
approved