0,3

Row sums are A000129. Diagonal sums are A008346. The Riordan array (1, s+tx) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).

Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1/2, -1/2, 0, 0, 0, 0, ...] DELTA [2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 10 2008

As an upper right triangle (in the example), table rows give number of points, edges, faces, cubes, 4D hypercubes etc. in hypercubes of increasing dimension by column. - Henry Bottomley, 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, May 08 2009

H. S. M. Coxeter, Regular Polytopes, Dover Publications, New York (1973), p. 122.

Eric W. Weisstein's Mathworld, Hypercube.

Number triangle T(n,k) = binomial(k, n-k)*2^k*(1/2)^(n-k); columns have g.f. (2*x+x^2)^k.

G.f.: 1/(1-2y*x-y*x^2). - Philippe Deléham, Nov 20 2011

Sum_ {k=0..n} T(n,k)*x^k = A000007(n), A000129(n+1), A090017(n+1), A090018(n), A190510(n+1), A190955(n+1) for x = 0,1,2,3,4,5 respectively. - Philippe Deléham, Nov 20 2011

T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = T(2,0) = 0, T(1,1) = 2, T(2,1) = 1, T(2,2) = 4, T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Oct 30 2013

Triangle begins:

1;

0, 2;

0, 1, 4;

0, 0, 4, 8;

0, 0, 1, 12, 16;

0, 0, 0, 6, 32, 32;

0, 0, 0, 1, 24, 80, 64;

The entries can also be interpreted as the antidiagonal reading of the following array:

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,... A000079

0, 1, 4, 12, 32, 80, 192, 448, 1024, 2304, 5120,... A001787

0, 0, 1, 6, 24, 80, 240, 672, 1792, 4608,11520,... A001788

0, 0, 0, 1, 8, 40, 160, 560, 1792, 5376,15360,... A001789

0, 0, 0, 0, 1, 10, 60, 280, 1120, 4032,13440,...

0, 0, 0, 0, 0, 1, 12, 84, 448, 2016, 8064,...

0, 0, 0, 0, 0, 0, 1, 14, 112, 672, 3360,...

0, 0, 0, 0, 0, 0, 0, 1, 16, 144, 960,...

0, 0, 0, 0, 0, 0, 0, 0, 1, 18, 180,...

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 20,...

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,...

Cf. A053118, A008312, A062715, A038207.

Sequence in context: A072737 A061290 A099096 * A121298 A372873 A212206

Adjacent sequences: A099086 A099087 A099088 * A099090 A099091 A099092

easy,nonn,tabl

Paul Barry, Sep 25 2004

approved