OFFSET
1,2
COMMENTS
Superfactorial A000178(n) equals the product of first n factorials.
FORMULA
a(n) ~ n^(n^2/2 + n + 17/12) * (2*Pi)^((n+1)/2) / (A * exp(3*n^2/4 + n - 1/12)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 10 2015
EXAMPLE
L.g.f.: L(x) = x + 3*x^2/2 + 31*x^3/3 + 1103*x^4/4 + 171311*x^5/5 +...
where
exp(L(x)) = 1 + x + 2*x^2 + 12*x^3 + 288*x^4 + 34560*x^5 + 24883200*x^6 + 125411328000*x^7 +...+ n!*(n-1)!*(n-2)!*...*3!*2!*1!*0!**x^n +...
MATHEMATICA
nmax=15; Rest[CoefficientList[Series[Log[Sum[BarnesG[k+2]*x^k, {k, 0, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]] (* Vaclav Kotesovec, Jul 10 2015 *)
PROG
(PARI) {a(n)=n*polcoeff(log(sum(k=0, n+1, prod(j=0, k, j!)*x^k)+x*O(x^n)), n)}
for(n=1, 21, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 16 2012
STATUS
approved