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 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 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

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