0,2

This infinite matrix is the square of the Pascal matrix (A007318) whose rows are [ 1,0,... ], [ 1,1,0,... ], [ 1,2,1,0,... ],...

As an upper right triangle, table rows give number of points, edges, faces, cubes, 4D hypercubes etc. in hypercubes of increasing dimension by column. - Henry Bottomley (se16(AT)btinternet.com), Apr 14 2000

Number of different partial sums of 1+[1,1,2]+[2,2,3]+[3,3,4]+[4,4,5]+... with entries that are zero removed. - Jon Perry (perry(AT)globalnet.co.uk), Jan 01 2004

Row sums are powers of 3 (A000244), antidiagonal sums are Pell numbers (A000129). - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), May 17 2005

Riordan array (1/(1-2x),x/(1-2x)). - Paul Barry (pbarry(AT)wit.ie), Jul 28 2005

T(n,k) is the number of elements of the Coxeter group B_n with descent set contained in {s_k}, 0<=k<=n-1. For T(n,n), we interpret this as the number of elements of B_n with empty descent set (since s_n does not exist). - Elizabeth Morris (epmorris(AT)math.washington.edu), Mar 01 2006

Let S be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xSy if x is a subset of y. Then T(n,k) = the number of elements (x,y) of S for which y has exactly k more elements than x. - Ross La Haye (rlahaye(AT)new.rr.com), Oct 12 2007

T(n,k) is number of paths in the first quadrant going from (0,0) to (n,k) using only steps B=(1,0) colored blue, R=(1,0) colored red, and U=(1,1). Example: T(3,2)=6 because we have BUU, RUU, UBU, URU, UUB, and UUR. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 04 2007

T(i,j) is the number of i-permutations of {1,2,3} containing j 1's. Example: T(2,1)=4 because we have 12, 13, 21, and 31; T(3,2)=6 because we have 112, 113, 121, 131, 211, and 311. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 21 2007

Triangle of coefficients in expansion of (2+x)^n. - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Apr 13 2008

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 155.

B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

S. J. Cyvin et al., Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.

W. G. Harter, Representations of multidimensional symmetries in networks, J. Math. Phys., 15 (1974), 2016-2021.

T. D. Noe, Rows n=0..100 of triangle, flattened

John Cartan, Starmaze: Cartan's Triangle.

T(n, k) = Sum[i=0..n, C(n, i)*C(i, k) ].

G.f.=1/(1-2z-tz). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 04 2007

Rows of the triangle are generated by taking successive iterates of (A135387)^n * [1, 0, 0, 0,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 09 2007

Triangle begins:

..................................1

.................................2, 1

...............................4, 4, 1

.............................8, 12, 6, 1

...........................16, 32, 24, 8, 1

........................32, 80, 80, 40, 10, 1

.....................64, 192, 240, 160, 60, 12, 1

..................128, 448, 672, 560, 280, 84, 14, 1

.............256, 1024, 1792, 1792, 1120, 448, 112, 16, 1

..........512, 2304, 4608, 5376, 4032, 2016, 672, 144, 18, 1

.....1024, 5120, 11520, 15360, 13440, 8064, 3360, 960, 180, 20, 1

..2048, 11264, 28160, 42240, 42240, 29568, 14784, 5280, 1320, 220,22, 1

4096, 24576, 67584, 112640, 126720, 101376, 59136, 25344, 7920,1760, 264, 24, 1

for i from 0 to 12 do seq(binomial(i, j)*2^(i-j), j = 0 .. i) end do; # yields sequence in t riangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 04 2007

(PARI) T(n, k)=polcoeff((x+2)^n, k) - Michael Somos, Apr 27 2000

(PARI) { n=13; v=vector(n); for (i=1, n, v[i]=vector(3^(i-1))); v[1][1]=1; for (i=2, n, k=length(v[i-1]); for (j=1, k, v[i][j]=v[i-1][j]+i; v[i][j+k]=v[i-1][j]+i; v[i][j+k+k]=v[i-1][j]+i+1)); c=vector(n); for (i=1, n, for (j=1, 3^(i-1), if (v[i][j]<=n, c[v[i][j]]++))); c } (Jon Perry)

Cf. A007318, A013609, A013610, etc. See also A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851, A062715.

Cf. A065109, A135387.

Apart from signs, same as A065109.

Adjacent sequences: A038204 A038205 A038206 this_sequence A038208 A038209 A038210

Sequence in context: A048807 A134395 A134397 this_sequence A065109 A113988 A134308

nonn,tabl,easy,nice,new

njas