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

0,5

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.

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.

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.

Example

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, ...].

Prog

(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(""))

Crossrefs

Cf. A323325, A165817 (row sums).

Sequence in context: A174388 A220718 A176283 * A181543 A141696 A178122

Adjacent sequences: A323321 A323322 A323323 * A323325 A323326 A323327

Keyword

nonn,tabl

Author

Paul D. Hanna, Jan 11 2019

Status

approved