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A207570 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(3*k-2) - 1).

%I A207570 #15 Feb 07 2020 13:37:15
%S A207570 1,1,4,34,410,6455,125251,2888305,77157780,2342972405,79701049425,
%T A207570 3002132647515,124039845584382,5577660227565634,271162541308698623,
%U A207570 14172237715785139175,792418822364402364530,47198077739119663907870,2983413619934353599892285
%N A207570 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(3*k-2) - 1).
%C A207570 Compare g.f. to: Sum_{n>=0} Product_{k=1..n} ((1+x)^k - 1), which is the g.f. of A179525.
%H A207570 Michael De Vlieger, <a href="/A207570/b207570.txt">Table of n, a(n) for n = 0..200</a>
%H A207570 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 A207570 a(n) ~ GAMMA(2/3) * 2^(2*n-1/3) * 3^(2*n+5/6) * n^(n-1/6) / (exp(n+Pi^2/72) * Pi^(2*n+7/6)). - _Vaclav Kotesovec_, May 06 2014
%e A207570 G.f.: A(x) = 1 + x + 4*x^2 + 34*x^3 + 410*x^4 + 6455*x^5 + 125251*x^6 +...
%e A207570 such that, by definition,
%e A207570 A(x) = 1 + ((1+x)-1) + ((1+x)-1)*((1+x)^4-1) + ((1+x)-1)*((1+x)^4-1)*((1+x)^7-1) + ((1+x)-1)*((1+x)^4-1)*((1+x)^7-1)*((1+x)^10-1) +...
%t A207570 Join[{1},Rest[With[{nn=20},CoefficientList[Series[Sum[Product[ (1+x)^(3k-2)-1,{k,n}],{n,nn}],{x,0,nn}],x]]]] (* _Harvey P. Dale_, Aug 20 2012 *)
%o A207570 (PARI) {a(n)=polcoeff(sum(m=0,n,prod(k=1,m,(1+x)^(3*k-2)-1) +x*O(x^n)),n)}
%o A207570 for(n=0,25,print1(a(n),", "))
%Y A207570 Cf. A179525, A207556, A207569, A207571.
%K A207570 nonn
%O A207570 0,3
%A A207570 _Paul D. Hanna_, Feb 18 2012