A207434 L.g.f.: log( Sum_{n>=0} Product_{k=1..n} ((1+x)^k - 1) ) = Sum_{n>=1} a(n)*x^n/n.

1, 3, 16, 103, 796, 7104, 71807, 810239, 10095145, 137686648, 2040943180, 32679948256, 562281127266, 10347659040127, 202849692259846, 4220573966037231, 92900793975348826, 2156973952747274733, 52686155932369860221, 1350605860832381895768, 36256679580764579284889

Offset: 1

Author

Paul D. Hanna, Feb 19 2012

Keywords

• nonn

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 16*x^3/3 + 103*x^4/4 + 796*x^5/5 + 7104*x^6/6 + ...
where exponentiation yields the g.f. of A179525:
exp(L(x)) = 1 + x + 2*x^2 + 7*x^3 + 33*x^4 + 197*x^5 + 1419*x^6 + 11966*x^7 + ...
such that, by definition,
exp(L(x)) = 1 + ((1+x)-1) + ((1+x)-1)*((1+x)^2-1) + ((1+x)-1)*((1+x)^2-1)*((1+x)^3-1) + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..250
• Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Crossrefs

Cf. A179525 (exp).

Programs

Mathematica

Rest@With[{m = 25}, CoefficientList[Series[Log[Sum[Product[(1+x)^k -1, {k, j}], {j,0,m+2}]], {x,0,m}], x]*Range[0, m]] (* G. C. Greubel, Feb 05 2020 *)

PARI

{a(n)=n*polcoeff(log(sum(m=0, n, prod(k=1, m, (1+x)^k-1, 1+x*O(x^n)))), n)}
for(n=1,31,print1(a(n),","))

Formula

L.g.f.: log( Sum_{n>=0} 1/(1+x)^(n^2) * Product_{k=1..n} ((1+x)^(2*k-1) - 1) ).