%I A038207
%S A038207 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,
%T A038207 12,1,128,448,672,560,280,84,14,1,256,1024,1792,1792,1120,448,112,16,1,
%U A038207 512,2304,4608,5376,4032,2016,672,144,18,1,1024,5120,11520,15360,13440
%N A038207 Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
%C A038207 This infinite matrix is the square of the Pascal matrix (A007318) whose 
               rows are [ 1,0,... ], [ 1,1,0,... ], [ 1,2,1,0,... ],...
%C A038207 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. 
               More precisely, the (i,j)-th entry is the number of j-dimensional 
               subspaces of an i-dimensional hypercube (see the Coxeter reference). 
               - Christof Weber (christof.weber(AT)igb.uzh.ch), May 08 2009
%C A038207 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
%C A038207 Row sums are powers of 3 (A000244), antidiagonal sums are Pell numbers 
               (A000129). - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), May 
               17 2005
%C A038207 Riordan array (1/(1-2x),x/(1-2x)). - Paul Barry (pbarry(AT)wit.ie), Jul 
               28 2005
%C A038207 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
%C A038207 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
%C A038207 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
%C A038207 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
%C A038207 Triangle of coefficients in expansion of (2+x)^n. - Nour-Eddine Fahssi 
               (fahssin(AT)yahoo.fr), Apr 13 2008
%C A038207 Sum of diagonals are Jacobsthal-numbers: A001045 [From M. Dols (markdols99(AT)yahoo.com), 
               Aug 31 2009]
%D A038207 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of 
               combinatorial proof, M.A.A. 2003, id. 155.
%D A038207 H. S. M. Coxeter, Regular Polytopes, Dover Publications, New York (1973), 
               p. 122.
%D A038207 B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal 
               systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 
               109-121.
%D A038207 S. J. Cyvin et al., Unbranched catacondensed polygonal systems containing 
               hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.
%D A038207 W. G. Harter, Representations of multidimensional symmetries in networks, 
               J. Math. Phys., 15 (1974), 2016-2021.
%D A038207 Ross La Haye, Binary Relations on the Power Set of an n-Element Set, 
               Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6. [From 
               Ross La Haye (rlahaye(AT)new.rr.com), Feb 22 2009]
%H A038207 T. D. Noe, <a href="b038207.txt">Rows n=0..100 of triangle, flattened</
               a>
%H A038207 John Cartan, <a href="http://www.cartania.com/starmaze/triangle.html">
               Starmaze: Cartan's Triangle</a>.
%F A038207 T(n, k) = Sum[i=0..n, C(n, i)*C(i, k) ].
%F A038207 G.f.=1/(1-2z-tz). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 04 
               2007
%F A038207 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
%F A038207 From the formalism of A133314, the e.g.f. for the row polynomials of 
               A038207 is exp(x*t)*exp(2x). The e.g.f. for the row polynomials of 
               the inverse matrix is exp(x*t)*exp(-2x). p iterates of the matrix 
               give the matrix with e.g.f. exp(x*t)*exp(p*2x). The results generalize 
               for 2 replaced by any number. [From Tom Copeland (tcjpn(AT)msn.com), 
               Aug 18 2008]
%e A038207 Triangle begins:
%e A038207 ...................................1
%e A038207 ..................................2, 1
%e A038207 ................................4, 4, 1
%e A038207 ..............................8, 12, 6, 1
%e A038207 ............................16, 32, 24, 8, 1
%e A038207 .........................32, 80, 80, 40, 10, 1
%e A038207 ......................64, 192, 240, 160, 60, 12, 1
%e A038207 ...................128, 448, 672, 560, 280, 84, 14, 1
%e A038207 ..............256, 1024, 1792, 1792, 1120, 448, 112, 16, 1
%e A038207 ...........512, 2304, 4608, 5376, 4032, 2016, 672, 144, 18, 1
%e A038207 ......1024, 5120, 11520, 15360, 13440, 8064, 3360, 960, 180, 20, 1
%e A038207 ...2048, 11264, 28160, 42240, 42240, 29568, 14784, 5280, 1320, 220,22, 
               1
%e A038207 .4096, 24576, 67584, 112640, 126720, 101376, 59136, 25344, 7920,1760, 
               264, 24, 1
%p A038207 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
%o A038207 (PARI) T(n,k)=polcoeff((x+2)^n,k) - Michael Somos, Apr 27 2000
%o A038207 (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)
%Y A038207 Cf. A007318, A013609, A013610, etc. See also A000079, A001787, A001788, 
               A001789, A003472, A054849, A002409, A054851, A062715.
%Y A038207 Cf. A065109, A135387.
%Y A038207 Apart from signs, same as A065109.
%Y A038207 Sequence in context: A048807 A134397 A134395 this_sequence A065109 A113988 
               A134308
%Y A038207 Adjacent sequences: A038204 A038205 A038206 this_sequence A038208 A038209 
               A038210
%K A038207 nonn,tabl,easy,nice
%O A038207 0,2
%A A038207 N. J. A. Sloane (njas(AT)research.att.com).