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

1, 1, 2, 8, 37, 219, 1557, 12994, 124427, 1344506, 16178891, 214522339, 3107144562, 48805300668, 826268787588, 14998055299920, 290550119360174, 5983278021430064, 130512410617529321, 3006012061455129053, 72900477505718600661

Offset: 0

Author

Peter Bala, Jul 24 2017

Keywords

• nonn
• easy

Comments

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.

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)).

Examples

			a(3) = 8: The eight row-Fishburn matrices of size 3 with odd nonzero entries are
(3) /1 1\
    \0 1/
/1 0 0\  /0 1 0\  /0 0 1\
|0 1 0|  |0 1 0|  |0 1 0|
\0 0 1/  \0 0 1/  \0 0 1/
/1 0 0\  /0 1 0\  /0 0 1\
|0 0 1|  |0 0 1|  |0 0 1|
\0 0 1/  \0 0 1/  \0 0 1/
		

References

• I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, p. 42.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..200
• Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Crossrefs

Cf. A158691, A179525, A289312, A289313, A289314, A289315, A289317.

Programs

Maple

C:= x -> x/(1 - x^2):
G:= add(mul( (1 + C(x))^k - 1, k=1..n),n=0..20):
S:= series(G,x,21):
seq(coeff(S,x,j),j=0..20);

Formula

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

a(n) ~ 12^(n+1) * n^(n + 1/2) / (exp(n + Pi^2/24) * Pi^(2*n + 3/2)). - Vaclav Kotesovec, Aug 31 2023