The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 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 distribution of color lattice points in the 765 r+g+b=k planes of 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 LINKS 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) 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[((1-x^n)/(1-x))^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: A160324 A347026 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 9 23:05 EST 2022. Contains 358710 sequences. (Running on oeis4.)