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A207433 G.f.: Sum_{n>=0} Product_{k=1..n} (q^k - 1) where q = (1-x^3)/(1-x).

%I A207433 #19 Feb 05 2020 23:52:28
%S A207433 1,1,3,11,56,350,2609,22582,222625,2462969,30219676,407276420,
%T A207433 5981197376,95073427910,1626294895274,29788176027819,581704672430937,
%U A207433 12064521684969823,264843222932272690,6135057298705027024,149559103545555671423,3827360866024134614644
%N A207433 G.f.: Sum_{n>=0} Product_{k=1..n} (q^k - 1) where q = (1-x^3)/(1-x).
%C A207433 Motivated by Peter Bala's identity described in A158690:
%C A207433 Sum_{n>=0} Product_{k=1..n} (q^k - 1) =
%C A207433 Sum_{n>=0} q^(-n^2) * Product_{k=1..n} (q^(2*k-1) - 1),
%C A207433 here q = (1+x)/(1+x^2). See cross-references for other examples.
%C A207433 At present Bala's identity is conjectural and needs formal proof.
%C A207433 a(n) = number of upper triangular matrices with entries from {0,1,2} with no zero rows such that the sum of the entries is n, that is, row Fishburn matrices of size n with entries from {0,1,2}. Cf. A179525. - _Peter Bala_, Nov 05 2017
%H A207433 Vaclav Kotesovec, <a href="/A207433/b207433.txt">Table of n, a(n) for n = 0..260</a>
%H A207433 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 A207433 G.f.: Sum_{n>=0} q^(-n^2) * Product_{k=1..n} (q^(2*k-1) - 1) where q = (1-x^3)/(1-x). [Based on Peter Bala's conjecture in A158690]
%F A207433 a(n) ~ exp(Pi^2/24) * 2^(2*n+3/2) * 3^(n+1) * n! / Pi^(2*n+2). - _Vaclav Kotesovec_, Aug 22 2017
%e A207433 G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 56*x^4 + 350*x^5 + 2609*x^6 +...
%e A207433 Let q = (1-x^3)/(1-x) = 1 + x + x^2, then
%e A207433 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 A207433 Also, we have the identity:
%e A207433 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 +...
%e A207433 From _Peter Bala_, Nov 05 2017: (Start)
%e A207433 a(3) = 11: The eleven row Fishburn matrices of size 3 with entries in {0,1,2} are
%e A207433 /1 0\ /2 0\ /0 1\ /0 2\ /1 1\
%e A207433 \0 2/ \0 1/ \0 2/ \0 1/ \0 1/
%e A207433 and
%e A207433 /1 0 0\ /0 1 0\ /0 0 1\ /1 0 0\ /0 1 0\ /0 0 1\
%e A207433 |0 1 0| |0 1 0| |0 0 1| |0 0 1| |0 0 1| |0 0 1|.
%e A207433 \0 0 1/ \0 0 1/ \0 0 1/ \0 0 1/ \0 0 1/ \0 0 1/
%e A207433 (End)
%o A207433 (PARI) {a(n)=local(A=1+x,q=(1+x+x^2 +x*O(x^n))); A=sum(m=0,n,prod(k=1,m,(q^k-1)));polcoeff(A,n)}
%o A207433 (PARI) {a(n)=local(A=1+x,q=(1+x+x^2 +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 A207433 for(n=0,25,print1(a(n),", "))
%Y A207433 Cf. A158690, A158691, A179525, A207386, A207397.
%K A207433 nonn,easy
%O A207433 0,3
%A A207433 _Paul D. Hanna_, Feb 17 2012