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%I #36 Aug 31 2023 05:10:05
%S 1,2,6,26,142,946,7446,67658,697118,8031586,102312486,1427905658,
%T 21666671534,355138949394,6253348428598,117720540700842,
%U 2359368991571518,50157679523340994,1127327559500923974,26709016625807923418,665292778385210384078
%N The number of upper-triangular matrices with integer entries whose absolute sum is equal to n, and each row and column contains at least one nonzero entry.
%C 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 column contains a nonzero entry. See A022493.
%C Here we consider generalized Fishburn matrices where we allow the Fishburn matrices to have positive and negative nonzero entries. We define the size of a generalized Fishburn matrix to be the absolute sum of the matrix entries. This sequence gives the number of generalized Fishburn matrices of size n.
%C Alternatively, this sequence gives the number of 2-colored Fishburn matrices of size n, that is, ordinary Fishburn matrices of size n where each nonzero entry in the matrix can have one of two different colors.
%C More generally, if F(x) = Sum_{n >= 0} (Product_{i = 1..n} 1 - 1/(1 + x)^i) is the o.g.f. for primitive Fishburn matrices A138265 (i.e., Fishburn matrices with entries restricted to the set {0,1}) and C(x) := c_1*x + c_2*x^2 + ..., where c_i is a sequence of nonnegative integers, then the composition F(C(x)) is the o.g.f. for colored Fishburn matrices where entry i in the matrix can have one of c_i different colors: c_i = 0 for some i means i does not appear as an entry in the Fishburn matrix. This result is an application of Lemma 2.2.22 of Goulden and Jackson.
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, p. 42.
%H Alois P. Heinz, <a href="/A289312/b289312.txt">Table of n, a(n) for n = 0..300</a>
%H Ankush Goswami, Abhash Kumar Jha, Byungchan Kim, and Robert Osburn, <a href="https://arxiv.org/abs/2204.02628">Asymptotics and sign patterns for coefficients in expansions of Habiro elements</a>, arXiv:2204.02628 [math.NT], 2022.
%H Hsien-Kuei Hwang and 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 G.f.: Sum_{n >= 0} Product_{k = 1..n} 1 - ((1 - x)/(1 + x))^k.
%F Alternative g.f.: Sum_{n >= 0} ((1 + x)/(1 - x))^(n+1) * Product_{k = 1..n} 1 - ((1 + x)/(1 - x))^k.
%F G.f.: B(2*x/(1+x)) where B(x) is the g.f. of A022493. - _Michael D. Weiner_, Feb 28 2019
%F a(n) ~ 2^(2*n + 5/2) * 3^(n + 3/2) * n^(n+1) / (exp(n) * Pi^(2*n+2)). - _Vaclav Kotesovec_, Aug 31 2023
%e a(2) = 6: The six upper triangular matrices of size 2 with no zero rows or columns are (+-2) and
%e /+-1 0\
%e | |.
%e \0 +-1/
%p G:= add(mul(1 - ((1-x)/(1+x))^k, k=1..n),n=0..20):
%p S:= series(G,x,21):
%p seq(coeff(S,x,j),j=0..20);
%p # _Peter Bala_, Jul 24 2017
%t m = 21; Sum[Product[1 - ((1-x)/(1+x))^k + O[x]^m, {k, 1, n}], {n, 0, m}] // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 28 2020 *)
%Y Cf. A022493, A138265, A289313.
%K nonn,easy
%O 0,2
%A _Peter Bala_, Jul 02 2017
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