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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 (list; graph; refs; listen; history; text; internal format)
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

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.

FORMULA

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

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);

CROSSREFS

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

Sequence in context: A020076 A342052 A317873 * A020130 A159051 A053520

Adjacent sequences:  A289313 A289314 A289315 * A289317 A289318 A289319

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Jul 24 2017

STATUS

approved

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Last modified August 17 23:13 EDT 2022. Contains 356204 sequences. (Running on oeis4.)