A109439 Triangle read by rows, in which row n gives coefficients in expansion of ((1 - x^n)/(1 - x))^3.

1, 1, 3, 3, 1, 1, 3, 6, 7, 6, 3, 1, 1, 3, 6, 10, 12, 12, 10, 6, 3, 1, 1, 3, 6, 10, 15, 18, 19, 18, 15, 10, 6, 3, 1, 1, 3, 6, 10, 15, 21, 25, 27, 27, 25, 21, 15, 10, 6, 3, 1, 1, 3, 6, 10, 15, 21, 28, 33, 36, 37, 36, 33, 28, 21, 15, 10, 6, 3, 1, 1, 3, 6, 10, 15, 21, 28, 36, 42, 46, 48, 48

Offset: 1

Author

Labos Elemer, Jun 30 2005

Keywords

• nonn
• tabf

Comments

Sum of n-th row is n^3. The n-th row contains 3n-2 entries. Largest coefficients in rows are listed in A077043. The 255th row describes the distribution of color lattice points in the 765 r+g+b=k planes of the 24-bit RGB-cube with 256^3 points.

Also, the number of cubes of dimension 1 X 1 X 1 needed to build a cube by layers perpendicular to the main diagonal. Each layer is made up of regular triangular numbers T near the summits and truncated T's in the middle. E.g., cube 3^3 is made of layers 1, 3, 6, 7, 6, 3, 1, using T1, T2, T3 and a regularly truncated T4, 7 instead of 10. - M. Dauchez (mdzzdm(AT)yahoo.fr), Aug 31 2005

The n-th row is the third row of the (n+1)-nomial triangle. For example, row 1 (1,3,3,1) is the third row in the binomial triangle; row 5 is the third row of the 6-nomial triangle. - Bob Selcoe, Feb 18 2014

It appears that T(n,k) gives the number of possible ways of randomly selecting k cards from n-1 sets, each with three different playing cards. - Juan Pablo Herrera P., Nov 04 2016

Examples

			Triangle starts:
  1;
  1, 3, 3, 1;
  1, 3, 6, 7, 6, 3, 1;
  1, 3, 6,10,12,12,10, 6, 3, 1;
  1, 3, 6,10,15,18,19,18,15,10, 6, 3, 1;
  1, 3, 6,10,15,21,25,27,27,25,21,15,10, 6, 3, 1;
  1, 3, 6,10,15,21,28,33,36,37,36,33,28,21,15,10, 6, 3, 1.
		

Crossrefs

Cf. A000217, A004737, A045943, A077043.

Programs

Mathematica

Flatten[Table[CoefficientList[Series[((1-x^n)/(1-x))^3,{x,1,3*n}],x], {n,1,100}],1]

PARI

row(n) = Vec(((1 - x^n)/(1 - x))^3);
tabf(nn) = for (n=1, nn, print(row(n))); \\ Michel Marcus, Oct 12 2016

Formula

From Juan Pablo Herrera P., Oct 17 2016: (Start)

T(n,k) = A000217(k+1) = (k+2)!/(k!*2) if 0 <= k < n,

T(n,k) = (9*n-3*n^2+6*k*n-6*k-2*k^2-4)/2 if n-3 < k < 2*n,

T(n,k) = A000217(3n-k-2) = (3*n-k-1)!/((3*(n-1)-k)!*2) if 2*n-3 < k < 3*n-2.

T(n,k) = Sum_{i=k-n+1..k} A004737(T(n,i)),

T(n,k) = Sum_{i=k-n+1..k} (n-|n-i-1|) if n <= k <= 2*n+1. (End)

Extensions

Offset corrected by Joerg Arndt at the suggestion of Michel Marcus, Oct 12 2016