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A207557 G.f.: Sum_{n>=0} 1/(1+x)^(n^2-n) * Product_{k=1..n} ((1+x)^(2*k-1) - 1).

%I A207557 #13 Nov 18 2019 05:35:29
%S A207557 1,1,3,12,64,420,3276,29581,303389,3483053,44245695,616103046,
%T A207557 9330961666,152700926414,2685132170466,50488787588936,
%U A207557 1010864433071206,21470488933116138,482176661100286182,11415700804801064258,284169548252819022230,7419733139418740010570
%N A207557 G.f.: Sum_{n>=0} 1/(1+x)^(n^2-n) * Product_{k=1..n} ((1+x)^(2*k-1) - 1).
%C A207557 Compare g.f. to: Sum_{n>=0} 1/(1+x)^(n^2) * Product_{k=1..n} ((1+x)^(2*k-1) - 1), which is the g.f. of A179525.
%H A207557 Vaclav Kotesovec, <a href="/A207557/b207557.txt">Table of n, a(n) for n = 0..135</a>
%H A207557 Hsien-Kuei Hwang, and 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], p. 36, 2019.
%F A207557 Given A(x) is the g.f. of this sequence, note that:
%F A207557 1 + x*A(x) = Sum_{n>=0} 1/(1+x)^(n^2+n) * Product_{k=1..n} ((1+x)^(2*k-1) - 1).
%F A207557 a(n) ~ 2*sqrt(6) * 12^(n+1) * n^(n+1) / (exp(n+Pi^2/24) * Pi^(2*n+3)). - _Vaclav Kotesovec_, May 07 2014
%e A207557 G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 64*x^4 + 420*x^5 + 3276*x^6 +...
%e A207557 such that, by definition,
%e A207557 A(x) = 1 + ((1+x)-1) + ((1+x)-1)*((1+x)^3-1)/(1+x)^2 + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1)/(1+x)^6 + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1)*((1+x)^7-1)/(1+x)^20 +...
%o A207557 (PARI) {a(n)=polcoeff(sum(m=0,n,prod(k=1,m,(1+x)^(2*k-1)-1)/(1+x+x*O(x^n))^(m^2-m) ),n)}
%o A207557 for(n=0,25,print1(a(n),", "))
%Y A207557 Cf. A179525, A207556.
%K A207557 nonn
%O A207557 0,3
%A A207557 _Paul D. Hanna_, Feb 18 2012