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A323324 Coefficients T(n,k) of x^n*y^(n-k)*z^k in function A = A(x,y,z) such that A = 1 + x*B*C, B = 1 + y*C*A, and C = 1 + z*A*B, as a triangle read by rows.

%I A323324 #24 Dec 09 2021 01:00:32
%S A323324 1,1,1,1,8,1,1,27,27,1,1,64,200,64,1,1,125,875,875,125,1,1,216,2835,
%T A323324 6272,2835,216,1,1,343,7546,30870,30870,7546,343,1,1,512,17472,118272,
%U A323324 217800,118272,17472,512,1,1,729,36450,378378,1146717,1146717,378378,36450,729,1,1,1000,70125,1056000,4879875,8016008,4879875,1056000,70125,1000,1,1,1331,126445,2647359,17649060,44088044,44088044,17649060,2647359,126445,1331,1,1,1728,216216,6086080,56119635,201636864,306330752,201636864,56119635,6086080,216216,1728,1
%N A323324 Coefficients T(n,k) of x^n*y^(n-k)*z^k in function A = A(x,y,z) such that A = 1 + x*B*C, B = 1 + y*C*A, and C = 1 + z*A*B, as a triangle read by rows.
%C A323324 Row sums equal A165817(n), the number of compositions of n into 2*n parts, for n >= 0.
%C A323324 Central terms equal 2*A165817(n)^2, for n >= 1.
%H A323324 Paul D. Hanna, <a href="/A323324/b323324.txt">Table of n, a(n) for n = 0..1325 terms of this triangle as read by rows 0..50</a>
%H A323324 Thomas Einolf, Robert Muth, and Jeffrey Wilkinson, <a href="https://arxiv.org/abs/2107.13417">Injectively k-colored rooted forests</a>, arXiv:2107.13417 [math.CO], 2021.
%F A323324 Sum_{k=0..n} T(n,k) = binomial(3*n-1, n) for n >= 0.
%F A323324 Sum_{k=0..n} k * T(n,k) = n * binomial(3*n-1, n-1), for n >= 0.
%F A323324 T(2*n,n) = 2 * binomial(3*n-1, n)^2 for n >= 1, with a(0) = 1.
%F A323324 T(n,k) = T(n,n-k) for k = 0..n, for n >= 0.
%F A323324 T(n,1) = n^3 for n >= 0.
%F A323324 T(n,2) = n^3*(n^2-1)*(2*n-3)/24 for n >= 0.
%e A323324 This triangle begins:
%e A323324 1;
%e A323324 1, 1;
%e A323324 1, 8, 1;
%e A323324 1, 27, 27, 1;
%e A323324 1, 64, 200, 64, 1;
%e A323324 1, 125, 875, 875, 125, 1;
%e A323324 1, 216, 2835, 6272, 2835, 216, 1;
%e A323324 1, 343, 7546, 30870, 30870, 7546, 343, 1;
%e A323324 1, 512, 17472, 118272, 217800, 118272, 17472, 512, 1;
%e A323324 1, 729, 36450, 378378, 1146717, 1146717, 378378, 36450, 729, 1;
%e A323324 1, 1000, 70125, 1056000, 4879875, 8016008, 4879875, 1056000, 70125, 1000, 1;
%e A323324 1, 1331, 126445, 2647359, 17649060, 44088044, 44088044, 17649060, 2647359, 126445, 1331, 1;
%e A323324 1, 1728, 216216, 6086080, 56119635, 201636864, 306330752, 201636864, 56119635, 6086080, 216216, 1728, 1; ...
%e A323324 ROW SUMS are
%e A323324 [1, 2, 10, 56, 330, 2002, 12376, 77520, 490314, ..., binomial(3*n-1, n), ...].
%e A323324 CENTRAL TERMS are
%e A323324 [1, 8, 200, 6272, 217800, 8016008, 306330752, ..., 2*binomial(3*n-1, n)^2, ...].
%o A323324 (PARI) {T(n,k) = my(A=1,B=1,C=1); for(i=0,n,
%o A323324 A = 1 + x*B*C +x*O(x^n);
%o A323324 B = 1 + y*A*C +y*O(y^n);
%o A323324 C = 1 + z*A*B +z*O(z^n));
%o A323324 polcoeff(polcoeff(polcoeff(A,n,x),n-k,y),k,z)}
%o A323324 for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))
%Y A323324 Cf. A323325, A165817 (row sums).
%K A323324 nonn,tabl
%O A323324 0,5
%A A323324 _Paul D. Hanna_, Jan 11 2019