OFFSET
0,2
FORMULA
a(n) = n!*Sum_{k=0..n} C(2*(n-k+1),k)/(n-k+1) * k^(n-k)/(n-k)!.
If A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then
a(n,m) = n!*Sum_{k=0..n} m*C(2*(n-k+m),k)/(n-k+m) * k^(n-k)/(n-k)!.
E.g.f.: A(x) = (1/x)*Series_Reversion(x/B(x)) where B(x) = (1 + x*exp(x)/B(x))^2.
a(n) ~ sqrt(2*s^(3/2)*(2-5*sqrt(s)+3*s)/(2*sqrt(s)-1)) * (2*s-2*sqrt(s))^n * n^(n-1) / exp(n), where s = 3.533778497303240223520495... is the root of the equation (2-2/sqrt(s)) * log(2*(sqrt(s)-2*s+s^(3/2))) = 1. - Vaclav Kotesovec, Jan 10 2014
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364978. - Seiichi Manyama, Nov 02 2024
EXAMPLE
E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 42*x^3/3! + 392*x^4/4! + 4970*x^5/5! +...
A(x)^(1/2) = 1 + x + 2*x^2/2! + 15*x^3/3! + 124*x^4/4! + 1565*x^5/5! +...
MATHEMATICA
Flatten[{1, Table[n!*Sum[Binomial[2*(n-k+1), k]/(n-k+1) * k^(n-k)/(n-k)!, {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jan 10 2014 *)
PROG
(PARI) {a(n, m=1)=n!*sum(k=0, n, m*binomial(2*(n-k+m), k)/(n-k+m)*k^(n-k)/(n-k)!)}
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Paul D. Hanna, Jun 18 2009
STATUS
approved