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.

1, 1, 1, 1, 8, 1, 1, 27, 27, 1, 1, 64, 200, 64, 1, 1, 125, 875, 875, 125, 1, 1, 216, 2835, 6272, 2835, 216, 1, 1, 343, 7546, 30870, 30870, 7546, 343, 1, 1, 512, 17472, 118272, 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

Offset: 0

Author

Paul D. Hanna, Jan 11 2019

Keywords

• nonn
• tabl

Comments

Row sums equal A165817(n), the number of compositions of n into 2*n parts, for n >= 0.

Central terms equal 2*A165817(n)^2, for n >= 1.

Examples

			This triangle begins:
1;
1, 1;
1, 8, 1;
1, 27, 27, 1;
1, 64, 200, 64, 1;
1, 125, 875, 875, 125, 1;
1, 216, 2835, 6272, 2835, 216, 1;
1, 343, 7546, 30870, 30870, 7546, 343, 1;
1, 512, 17472, 118272, 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; ...
ROW SUMS are
[1, 2, 10, 56, 330, 2002, 12376, 77520, 490314, ..., binomial(3*n-1, n), ...].
CENTRAL TERMS are
[1, 8, 200, 6272, 217800, 8016008, 306330752, ..., 2*binomial(3*n-1, n)^2, ...].
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1325 terms of this triangle as read by rows 0..50
• Thomas Einolf, Robert Muth, and Jeffrey Wilkinson, Injectively k-colored rooted forests, arXiv:2107.13417 [math.CO], 2021.

Crossrefs

Cf. A323325, A165817 (row sums).

Programs

PARI

{T(n,k) = my(A=1,B=1,C=1); for(i=0,n,
A = 1 + x*B*C +x*O(x^n);
B = 1 + y*A*C +y*O(y^n);
C = 1 + z*A*B +z*O(z^n));
polcoeff(polcoeff(polcoeff(A,n,x),n-k,y),k,z)}
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))

Formula

Sum_{k=0..n} T(n,k) = binomial(3*n-1, n) for n >= 0.

Sum_{k=0..n} k * T(n,k) = n * binomial(3*n-1, n-1), for n >= 0.

T(2*n,n) = 2 * binomial(3*n-1, n)^2 for n >= 1, with a(0) = 1.

T(n,k) = T(n,n-k) for k = 0..n, for n >= 0.

T(n,1) = n^3 for n >= 0.

T(n,2) = n^3*(n^2-1)*(2*n-3)/24 for n >= 0.