login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A099089 Riordan array (1, 2+x). 9
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, 0, 0, 0, 0, 8, 80, 192, 128, 0, 0, 0, 0, 1, 40, 240, 448, 256, 0, 0, 0, 0, 0, 10, 160, 672, 1024, 512, 0, 0, 0, 0, 0, 1, 60, 560, 1792, 2304, 1024, 0, 0, 0, 0, 0, 0, 12, 280, 1792, 4608, 5120, 2048 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

REFERENCES

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

LINKS

Table of n, a(n) for n=0..77.

Eric W. Weisstein's Mathworld, Hypercube.

FORMULA

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

EXAMPLE

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,...

CROSSREFS

Cf. A053118, A008312, A062715, A038207.

Sequence in context: A072737 A061290 A099096 * A121298 A212206 A247489

Adjacent sequences:  A099086 A099087 A099088 * A099090 A099091 A099092

KEYWORD

easy,nonn,tabl,changed

AUTHOR

Paul Barry, Sep 25 2004

STATUS

approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 10 02:20 EDT 2020. Contains 333392 sequences. (Running on oeis4.)