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A289316 The number of upper-triangular matrices whose nonzero entries are positive odd numbers summing to n and each row contains a nonzero entry.

%I A289316 #14 Aug 31 2023 04:51:12
%S A289316 1,1,2,8,37,219,1557,12994,124427,1344506,16178891,214522339,
%T A289316 3107144562,48805300668,826268787588,14998055299920,290550119360174,
%U A289316 5983278021430064,130512410617529321,3006012061455129053,72900477505718600661
%N A289316 The number of upper-triangular matrices whose nonzero entries are positive odd numbers summing to n and each row contains a nonzero entry.
%C A289316 A row-Fishburn matrix of size n is defined to be an upper-triangular matrix with nonnegative integer entries which sum to n and each row contains a nonzero entry. See A158691. Here we are considering row-Fishburn matrices where the nonzero entries are all odd.
%C A289316 The g.f. F(x) for primitive row_Fishburn matrices (i.e., row_Fishburn matrices with entries restricted to the set {0,1}), is F(x) = Sum_{n>=0} Product_{k=1..n} ( (1 + x)^k - 1 ). See A179525. Let C(x) = x/(1 - x^2) = x + x^3 + x^5 + x^7 + .... Then appplying Lemma 2.2.22 of Goulden and Jackson gives the g.f. for the present sequence as the composition F(C(x)).
%D A289316 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, p. 42.
%H A289316 Seiichi Manyama, <a href="/A289316/b289316.txt">Table of n, a(n) for n = 0..200</a>
%H A289316 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 A289316 G.f.: A(x) = Sum_{n >= 0} Product_{k = 1..n} ( (1 + x/(1 - x^2))^k - 1 ).
%F A289316 a(n) ~ 12^(n+1) * n^(n + 1/2) / (exp(n + Pi^2/24) * Pi^(2*n + 3/2)). - _Vaclav Kotesovec_, Aug 31 2023
%e A289316 a(3) = 8: The eight row-Fishburn matrices of size 3 with odd nonzero entries are
%e A289316 (3) /1 1\
%e A289316     \0 1/
%e A289316 /1 0 0\  /0 1 0\  /0 0 1\
%e A289316 |0 1 0|  |0 1 0|  |0 1 0|
%e A289316 \0 0 1/  \0 0 1/  \0 0 1/
%e A289316 /1 0 0\  /0 1 0\  /0 0 1\
%e A289316 |0 0 1|  |0 0 1|  |0 0 1|
%e A289316 \0 0 1/  \0 0 1/  \0 0 1/
%p A289316 C:= x -> x/(1 - x^2):
%p A289316 G:= add(mul( (1 + C(x))^k - 1, k=1..n),n=0..20):
%p A289316 S:= series(G,x,21):
%p A289316 seq(coeff(S,x,j),j=0..20);
%Y A289316 Cf. A158691, A179525, A289312, A289313, A289314, A289315, A289317.
%K A289316 nonn,easy
%O A289316 0,3
%A A289316 _Peter Bala_, Jul 24 2017