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A207386 G.f.: Sum_{n>=0} Product_{k=1..n} (q^k - 1) where q = (1+x)/(1+x^3).

%I A207386 #16 Feb 05 2020 23:52:16
%S A207386 1,1,2,6,28,172,1269,10879,106343,1167970,14241792,190919195,
%T A207386 2790920003,44184957237,753152722642,13752229833566,267809474619299,
%U A207386 5540559819166056,121355678158129804,2805498395990301867,68265999939081386947,1744058001878302097109
%N A207386 G.f.: Sum_{n>=0} Product_{k=1..n} (q^k - 1) where q = (1+x)/(1+x^3).
%C A207386 Motivated by Peter Bala's identity described in A158690:
%C A207386 Sum_{n>=0} Product_{k=1..n} (q^k - 1) =
%C A207386 Sum_{n>=0} q^(-n^2) * Product_{k=1..n} (q^(2*k-1) - 1),
%C A207386 here q = (1+x)/(1+x^3). See cross-references for other examples.
%C A207386 At present Bala's identity is conjectural and needs formal proof.
%H A207386 Vaclav Kotesovec, <a href="/A207386/b207386.txt">Table of n, a(n) for n = 0..250</a>
%H A207386 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 A207386 G.f.: Sum_{n>=0} q^(-n^2) * Product_{k=1..n} (q^(2*k-1) - 1) where q = (1+x)/(1+x^3). [Based on Peter Bala's conjecture in A158690]
%F A207386 a(n) ~ n! * 2^(2*n+3/2) * 3^(n+1) / (exp(Pi^2/24) * Pi^(2*n+2)). - _Vaclav Kotesovec_, Aug 22 2017
%e A207386 G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 28*x^4 + 172*x^5 + 1269*x^6 +...
%e A207386 Let q = (1+x)/(1+x^3) = 1/(1-x+x^2), then
%e A207386 A(x) = 1 + (q-1) + (q-1)*(q^2-1) + (q-1)*(q^2-1)*(q^3-1) + (q-1)*(q^2-1)*(q^3-1)*(q^4-1) + (q-1)*(q^2-1)*(q^3-1)*(q^4-1)*(q^5-1) +...
%e A207386 which also is proposed to equal:
%e A207386 A(x) = 1 + (q-1)/q + (q-1)*(q^3-1)/q^4 + (q-1)*(q^3-1)*(q^5-1)/q^9 + (q-1)*(q^3-1)*(q^5-1)*(q^7-1)/q^16 + (q-1)*(q^3-1)*(q^5-1)*(q^7-1)*(q^9-1)/q^25 +...
%o A207386 (PARI) {a(n)=local(A=1+x,q=(1+x)/(1+x^3 +x*O(x^n))); A=sum(m=0,n,prod(k=1,m,(q^k-1)));polcoeff(A,n)}
%o A207386 (PARI) {a(n)=local(A=1+x,q=(1+x)/(1+x^3 +x*O(x^n))); A=sum(m=0,n,q^(-m^2)*prod(k=1,m,(q^(2*k-1)-1)));polcoeff(A,n)}
%o A207386 for(n=0,21,print1(a(n),", "))
%Y A207386 Cf. A158690, A158691, A179525, A207397, A207433.
%K A207386 nonn
%O A207386 0,3
%A A207386 _Paul D. Hanna_, Feb 17 2012