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A099089
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Riordan array (1, 2+x).
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9
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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
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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H. S. M. Coxeter, Regular Polytopes, Dover Publications, New York (1973), p. 122.
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LINKS
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FORMULA
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Number triangle T(n,k) = binomial(k, n-k)*2^k*(1/2)^(n-k); columns have g.f. (2*x+x^2)^k.
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
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EXAMPLE
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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,...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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