cp's OEIS Frontend

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

%I A207434 #18 Feb 05 2020 23:52:35
%S A207434 1,3,16,103,796,7104,71807,810239,10095145,137686648,2040943180,
%T A207434 32679948256,562281127266,10347659040127,202849692259846,
%U A207434 4220573966037231,92900793975348826,2156973952747274733,52686155932369860221,1350605860832381895768,36256679580764579284889
%N A207434 L.g.f.: log( Sum_{n>=0} Product_{k=1..n} ((1+x)^k - 1) ) = Sum_{n>=1} a(n)*x^n/n.
%H A207434 G. C. Greubel, <a href="/A207434/b207434.txt">Table of n, a(n) for n = 1..250</a>
%H A207434 Hsien-Kuei Hwang, Emma Yu Jin, <a href="https://arxiv.org/abs/1911.06690">Asymptotics and statistics on Fishburn matrices and their generalizations</a>, arXiv:1911.06690 [math.CO], 2019.
%F A207434 L.g.f.: log( Sum_{n>=0} 1/(1+x)^(n^2) * Product_{k=1..n} ((1+x)^(2*k-1) - 1) ).
%e A207434 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 + ...
%e A207434 where exponentiation yields the g.f. of A179525:
%e A207434 exp(L(x)) = 1 + x + 2*x^2 + 7*x^3 + 33*x^4 + 197*x^5 + 1419*x^6 + 11966*x^7 + ...
%e A207434 such that, by definition,
%e A207434 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) + ...
%t A207434 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 *)
%o A207434 (PARI) {a(n)=n*polcoeff(log(sum(m=0, n, prod(k=1, m, (1+x)^k-1, 1+x*O(x^n)))), n)}
%o A207434 for(n=1,31,print1(a(n),","))
%Y A207434 Cf. A179525 (exp).
%K A207434 nonn
%O A207434 1,2
%A A207434 _Paul D. Hanna_, Feb 19 2012