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%I #15 Feb 07 2020 13:38:07
%S 1,1,3,10,45,249,1709,13912,131168,1402706,16757321,221018769,
%T 3188425939,49925523804,843121969923,15272776193787,295372123082865,
%U 6073931908657770,132329525329523223,3044691799670213778,73771773281455834427,1877511491197391256001
%N G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^k - 1)/(1 - x^k).
%H Vaclav Kotesovec, <a href="/A207652/b207652.txt">Table of n, a(n) for n = 0..210</a>
%H 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 From _Vaclav Kotesovec_, Oct 31 2014: (Start)
%F a(n) ~ 6*sqrt(2) * 12^n * n! / (exp(Pi^2/24) * Pi^(2*n+2)).
%F a(n) ~ 12^(n+1) * n^(n+1/2) / (exp(n + Pi^2/24) * Pi^(2*n+3/2)).
%F (End)
%e G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 45*x^4 + 249*x^5 + 1709*x^6 +...
%e such that, by definition,
%e A(x) = 1 + ((1+x)-1)/(1-x) + ((1+x)-1)*((1+x)^2-1)/((1-x)*(1-x^2)) + ((1+x)-1)*((1+x)^2-1)*((1+x)^3-1)/((1-x)*(1-x^2)*(1-x^3)) +...
%o (PARI) {a(n)=polcoeff(sum(m=0,n,prod(k=1,m,((1+x)^k-1)/(1-x^k +x*O(x^n)) )),n)}
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A207651, A207653, A207654.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 19 2012
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