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A289317 The number of upper-triangular matrices whose nonzero entries are positive odd numbers summing to n and each row and each column contains a nonzero entry.

%I A289317 #15 Aug 31 2023 05:01:36
%S A289317 1,1,1,3,7,23,84,364,1792,9953,61455,417720,3098515,24902930,
%T A289317 215538825,1998518430,19761943208,207571259703,2307812703419,
%U A289317 27075591512866,334263981931669
%N A289317 The number of upper-triangular matrices whose nonzero entries are positive odd numbers summing to n and each row and each column contains a nonzero entry.
%C A289317 A Fishburn matrix of size n is defined to be an upper-triangular matrix with nonnegative integer entries which sum to n and each row and each column contains a nonzero entry. See A022493. Here we are considering Fishburn matrices where the nonzero entries are all odd.
%C A289317 The g.f. for primitive Fishburn matrices (i.e., Fishburn matrices with entries restricted to the set {0,1}), is F(x) = Sum_{n>=0} Product_{k=1..n} ( 1 - 1/(1 + x)^k ). See A138265. Let C(x) = x/(1 - x^2) = x + x^3 + x^5 + x^7 + .... Then applying Lemma 2.2.22 of Goulden and Jackson gives the g.f. for this sequence as the composition F(C(x)).
%D A289317 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, p. 42.
%H A289317 Seiichi Manyama, <a href="/A289317/b289317.txt">Table of n, a(n) for n = 0..200</a>
%H A289317 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 A289317 G.f.: A(x) = Sum_{n >= 0} Product_{k = 1..n} ( 1 - 1/(1 + x/(1 - x^2))^k ).
%F A289317 a(n) ~ 2^(n + 5/2) * 3^(n + 3/2) * n^(n+1) / (exp(n + Pi^2/12) * Pi^(2*n + 2)). - _Vaclav Kotesovec_, Aug 31 2023
%e A289317 a(4) = 7: The Fishburn matrices of size 4 with odd nonzero entries are
%e A289317 /3 0\ /1 0\
%e A289317 \0 1/ \0 3/
%e A289317 /1 1 0\ /1 0 1\ /1 0 0\
%e A289317 |0 1 0| |0 1 0| |0 1 1|
%e A289317 \0 0 1/ \0 0 1/ \0 0 1/
%e A289317 /1 1 0\
%e A289317 |0 0 1|
%e A289317 \0 0 1/
%e A289317 /1 0 0 0\
%e A289317 |0 1 0 0|
%e A289317 |0 0 1 0|
%e A289317 \0 0 0 1/
%p A289317 C:= x -> x/(1 - x^2):
%p A289317 G:= add(mul( 1 - 1/(1 + C(x))^k, k=1..n), n=0..20):
%p A289317 S:= series(G,x,21):
%p A289317 seq(coeff(S,x,j),j=0..20);
%Y A289317 Cf. A022493, A138265, A289312, A289313, A289314, A289315, A289316.
%K A289317 nonn,easy
%O A289317 0,4
%A A289317 _Peter Bala_, Jul 25 2017