OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..144
FORMULA
a(n) ~ exp(1/2) * d^(n+1) * (n!)^2, where d = 1/(Ei(1)-gamma) = 1/(A091725 - A001620) = 0.75878167350772..., where Ei is the second exponential integral and gamma is the Euler-Mascheroni constant. - Vaclav Kotesovec, Nov 02 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 20*x^3/3! + 242*x^4/4! + 4584*x^5/5! + ...
where
A(x) = 1 + log(1+x) + log(1+x)*log(1+2*x) + log(1+x)*log(1+2*x)*log(1+3*x) + log(1+x)*log(1+2*x)*log(1+3*x)*log(1+4*x) + ...
MAPLE
a:=series(add(mul(log(1+k*x), k=1..n), n=0..100), x=0, 18): seq(n!*coeff(a, x, n), n=0..17); # Paolo P. Lava, Mar 27 2019
MATHEMATICA
With[{nmax = 30}, CoefficientList[Series[Sum[Product[Log[1 + j*x], {j, 1, k}], {k, 0, 3*nmax}], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Sep 04 2018 *)
PROG
(PARI) {a(n)=n!*polcoeff(sum(m=0, n, prod(k=1, m, log(1+k*x+x*O(x^n)))), n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 22 2012
STATUS
approved