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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.
2
1, 1, 2, 8, 37, 219, 1557, 12994, 124427, 1344506, 16178891, 214522339, 3107144562, 48805300668, 826268787588, 14998055299920, 290550119360174, 5983278021430064, 130512410617529321, 3006012061455129053, 72900477505718600661
OFFSET
0,3
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)).
REFERENCES
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, p. 42.
LINKS
Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.
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
EXAMPLE
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/
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);
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Jul 24 2017
STATUS
approved