0,3

Also sum of rows in A046816. - Lior Manor (lior.manor(AT)gmail.com) Apr 24 2004

Also square array of unsigned coefficients of Chebyshev polynomials of second kind . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 12 2005

The rows give the number of k-simplices in the n-cube. For example, 1, 6, 12, 8 shows that the 3-cube has 1 volume, 6 faces, 12 edges and 8 vertices. - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Jun 05 2006

Triangle whose (i, j)-th entry is binomial(i, j)*2^j.

With offset [1,1] the triangle with doubled numbers, 2*a(n,m), enumerates sequences of length m with nonzero integer entries n_i satisfying sum(|n_i|)<=n. Example n=4, m=2: [1,3], [3,1], [2,2] each in 2^2=4 signed versions: 2*a(4,2)=2*6=12. The Sum over m (row sums of 2*a(n,m)) gives 2*3^(n-1), n>=1. See the W. Lang comment and a K. A. Meissner reference under A024023. - W. Lang, Jan 21 2008.

n-th row of the triangle = leftmost column of nonzero terms of X^n, where X = an infinite bidiagonal matrix with (1,1,1,...) in the main diagonal and (2,2,2,...) in the subdiagonal. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 19 2008

Numerators of a matrix square-root of Pascal's triangle A007318, where the denominators for the n-th row are set to 2^n. [From Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Aug 20 2009]

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

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

T. D. Noe, Rows n=0..50 of triangle, flattened

John Cartan, Cartan's triangle shows the relationship to the n-cube.

G.f.: 1 / [1 - x(1+2y)].

bin2(n, k) = 2*bin2(n-1, k-1) + bin2(n-1, k) (e.g. 1, 4, 4 gives 1, 2.1+4=6, 2.4+4=8 and 2.4=8) - Jon Perry (perry(AT)globalnet.co.uk), Nov 22 2005

Triangle begins:

.1,

.1,2,

.1,4,4,

.1,6,12,8,

.1,8,24,32,16,

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

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

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

....

bin2:=proc(n, k) option remember; if k<0 or k>n then 0 elif k=0 then 1 else 2*bin2(n-1, k-1)+bin2(n-1, k); fi; end; # N. J. A. Sloane, Jun 01 2009

Cf. A007318, A013610, etc.

Row sums are in A000244.

Sequence in context: A136672 A097750 A133544 this_sequence A154558 A008572 A118976

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 23 2009: (Start)

Appears in A167580 and A167591.

(End)

Adjacent sequences: A013606 A013607 A013608 this_sequence A013610 A013611 A013612

tabl,nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).