A219541 Expansion of e.g.f.: Sum_{n>=0} Product_{k=1..n} log(1 + k*x).

1, 1, 3, 20, 242, 4584, 124936, 4638360, 225037200, 13820428368, 1048006461024, 96171381464256, 10503700943629824, 1346451508974957696, 200184649396819872768, 34167655864475762390784, 6635466680845611611326464, 1454780635849943337186155520

Offset: 0

Author

Paul D. Hanna, Nov 22 2012

Keywords

• nonn

Examples

			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) + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..144

Crossrefs

Cf. A091725, A001620.

Programs

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 *)

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),", "))

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