OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..160
FORMULA
a(n) = Sum_{k=0..n} k^(2*n) * k! * Stirling2(n, k).
E.g.f.: Sum_{n>=0} (exp(n^2*x) - 1)^n.
a(n) ~ c * d^n * (n!)^3 / n, where d = r^3*(1+exp(2/r)) = 7.8512435106631367719817991716164612615296980032514..., r = 0.94520217245242431308104743874492469552738... is the root of the equation (1+exp(-2/r))*LambertW(-exp(-1/r)/r) = -1/r, and c = 0.142680262107781025906560380273234930916319644... . - Vaclav Kotesovec, May 08 2014
EXAMPLE
O.g.f.: A(x) = 1 + x + 33*x^2 + 4759*x^3 + 1812645*x^4 + 1432421311*x^5 +...
where
A(x) = 1 + x/(1-x) + 2!*2^4*x^2/((1-2^2*1*x)*(1-2^2*2*x)) + 3!*3^6*x^3/((1-3^2*1*x)*(1-3^2*2*x)*(1-3^2*3*x)) + 4!*4^8*x^4/((1-4^2*1*x)*(1-4^2*2*x)*(1-4^2*3*x)*(1-4^2*4*x)) +...
Exponential Generating Function.
E.g.f.: E(x) = 1 + x + 33*x^2/2! + 4759*x^3/3! + 1812645*x^4/4! +...
where
E(x) = 1 + (exp(x)-1) + (exp(4*x)-1)^2 + (exp(9*x)-1)^3 + (exp(16*x)-1)^4 + (exp(25*x)-1)^5 + (exp(36*x)-1)^6 + (exp(49*x)-1)^7 +...
MATHEMATICA
Flatten[{1, Table[Sum[k^(2*n) * k! * StirlingS2[n, k], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 08 2014 *)
PROG
(PARI) {a(n)=polcoeff(sum(m=0, n, m!*m^(2*m)*x^m/prod(k=1, m, 1-m^2*k*x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=n!*polcoeff(sum(m=0, n, (exp(m^2*x+x*O(x^n))-1)^m), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=sum(k=0, n, k^(2*n) * k! * Stirling2(n, k))}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 17 2013
STATUS
approved