OFFSET
0,4
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..380
FORMULA
a(n) = n!*Sum_{k=0..n} C(n-k+1,k)/(n-k+1) * Sum_{j=0..k} C(k,j)/2^k*(2j-k)^(n-k)/(n-k)!.
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = n!*Sum_{k=0..n} m*C(n-k+m,k)/(n-k+m) * Sum_{j=0..k} C(k,j)/2^k*(2j-k)^(n-k)/(n-k)!.
a(n) ~ n^(n-1) * sqrt((2*s-1)/(s-1)) / (exp(n) * r^(n+1)), where r = 0.6184142504137720756... and s = 2.731257206829781545... are roots of the system of equations r^2*sinh(r*s) = 1, s = 1 + r*cosh(r*s). - Vaclav Kotesovec, Jul 15 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^3/3! + 24*x^4/4! + 65*x^5/5! + 480*x^6/6! +...
PROG
(PARI) {a(n, m=1)=n!*sum(k=0, n, m*binomial(n-k+m, k)/(n-k+m)*sum(j=0, k, binomial(k, j)/2^k*(2*j-k)^(n-k)/(n-k)!))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 09 2009
STATUS
approved