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A109439 Triangle read by rows, in which row n gives coefficients in expansion of ((1 - x^n)/(1 - x))^3. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Sum of n-th row is (n+1)^3. The n-th row contains 3n+1 entries. Largest coefficients in rows are listed in A077043. The 255-th row describes distribution of color lattice points in the 765 r+g+b=k planes of 24 bitRGB-cube with 256^3 points.

The number of cubes 1 X 1 X 1 used to build a cube by layers perpendicular to the great diagonal. Each layer is made 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

LINKS

Table of n, a(n) for n=0..81.

EXAMPLE

The 0-th to 6th rows are

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.

Sum of 5-th row is 216=(5+1)^3=216.

MATHEMATICA

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

CROSSREFS

Cf. A000217, A077043, A045943.

Sequence in context: A090569 A160324 A197928 * A247646 A133333 A171876

Adjacent sequences:  A109436 A109437 A109438 * A109440 A109441 A109442

KEYWORD

nonn,tabf

AUTHOR

Labos Elemer, Jun 30 2005

STATUS

approved

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Last modified December 19 19:14 EST 2014. Contains 252239 sequences.