

A109439


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


6



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
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OFFSET

1,3


COMMENTS

Sum of nth row is n^3. The nth row contains 3n2 entries. Largest coefficients in rows are listed in A077043. The 255th row describes distribution of color lattice points in the 765 r+g+b=k planes of 24bit RGBcube 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 nth 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 6nomial triangle.  Bob Selcoe, Feb 18 2014
It appears that T(n,k) gives the number of possible ways of randomly selecting k cards from n1 sets, each with three different playing cards.  Juan Pablo Herrera P., Nov 04 2016


LINKS

Table of n, a(n) for n=1..82.


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*n3*n^2+6*k*n6*k2*k^24)/2 if n3 < k < 2*n,
T(n,k) = A000217(3nk2) = (3*nk1)!/((3*(n1)k)!*2) if 2*n3 < k < 3*n2.
T(n,k) = Sum_{i=kn+1..k} A004737(T(n,i)),
T(n,k) = Sum_{i=kn+1..k} (nni1) if n <= k <= 2*n+1. (End)


EXAMPLE

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.


MATHEMATICA

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


PROG

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


CROSSREFS

Cf. A000217, A004737, A045943, A077043.
Sequence in context: A090569 A160324 A197928 * A247646 A133333 A296523
Adjacent sequences: A109436 A109437 A109438 * A109440 A109441 A109442


KEYWORD

nonn,tabf


AUTHOR

Labos Elemer, Jun 30 2005


EXTENSIONS

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


STATUS

approved



