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
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.
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.
Comments