OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..212
FORMULA
Sum_{n>=0} a(n)*x^n / n!^2 = Sum_{n>=0} (exp(n*x) - 1)^n / n!^2.
a(n) = n! * Sum_{k=0..n} k^n * Stirling2(n,k) / k!.
EXAMPLE
E.g.f.: A(x) = 1 + x + 6*x^2/2! + 105*x^3/3! + 3568*x^4/4! + 204745*x^5/5! +...
where
A(x) = 1 + x/(1-x) + 2^2*x^2/(2!*(1-2*1*x)*(1-2*2*x)) + 3^3*x^3/(3!*(1-3*1*x)*(1-3*2*x)*(1-3*3*x)) + 4^4*x^4/(4!*(1-4*1*x)*(1-4*2*x)*(1-4*3*x)*(1-4*4*x)) +...
MATHEMATICA
Flatten[{1, Table[n! * Sum[k^n * StirlingS2[n, k] / k!, {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 08 2014 *)
PROG
(PARI) {a(n)=n!*polcoeff(sum(m=0, 20, m^m*x^m/m!/prod(k=1, m, 1-m*k*x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=n!^2*polcoeff(sum(m=0, n, (exp(m*x+x*O(x^n))-1)^m/m!^2), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=n!*sum(k=0, n, k^n*stirling(n, k, 2)/k!)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 18 2013
STATUS
approved