|
%I #30 Apr 10 2020 02:07:51
%S 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,
%T 0,8,80,192,128,0,0,0,0,1,40,240,448,256,0,0,0,0,0,10,160,672,1024,
%U 512,0,0,0,0,0,1,60,560,1792,2304,1024,0,0,0,0,0,0,12,280,1792,4608,5120,2048
%N Riordan array (1, 2+x).
%C 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).
%C 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
%C 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
%D H. S. M. Coxeter, Regular Polytopes, Dover Publications, New York (1973), p. 122.
%H Eric W. Weisstein's Mathworld, <a href="http://mathworld.wolfram.com/Hypercube.html">Hypercube</a>.
%F Number triangle T(n,k) = binomial(k, n-k)*2^k*(1/2)^(n-k); columns have g.f. (2*x+x^2)^k.
%F G.f.: 1/(1-2y*x-y*x^2). - _Philippe Deléham_, Nov 20 2011
%F 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
%F 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
%e Triangle begins:
%e 1;
%e 0, 2;
%e 0, 1, 4;
%e 0, 0, 4, 8;
%e 0, 0, 1, 12, 16;
%e 0, 0, 0, 6, 32, 32;
%e 0, 0, 0, 1, 24, 80, 64;
%e The entries can also be interpreted as the antidiagonal reading of the following array:
%e 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,... A000079
%e 0, 1, 4, 12, 32, 80, 192, 448, 1024, 2304, 5120,... A001787
%e 0, 0, 1, 6, 24, 80, 240, 672, 1792, 4608,11520,... A001788
%e 0, 0, 0, 1, 8, 40, 160, 560, 1792, 5376,15360,... A001789
%e 0, 0, 0, 0, 1, 10, 60, 280, 1120, 4032,13440,...
%e 0, 0, 0, 0, 0, 1, 12, 84, 448, 2016, 8064,...
%e 0, 0, 0, 0, 0, 0, 1, 14, 112, 672, 3360,...
%e 0, 0, 0, 0, 0, 0, 0, 1, 16, 144, 960,...
%e 0, 0, 0, 0, 0, 0, 0, 0, 1, 18, 180,...
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 20,...
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,...
%Y Cf. A053118, A008312, A062715, A038207.
%K easy,nonn,tabl
%O 0,3
%A _Paul Barry_, Sep 25 2004
|
