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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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

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

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Last modified November 24 20:34 EST 2020. Contains 338616 sequences. (Running on oeis4.)