%I A008949
%S A008949 1,1,2,1,3,4,1,4,7,8,1,5,11,15,16,1,6,16,26,31,32,1,7,22,42,57,63,64,1,
%T A008949 8,29,64,99,120,127,128,1,9,37,93,163,219,247,255,256,1,10,46,130,256,
%U A008949 382,466,502,511,512,1,11,56,176,386,638,848,968,1013,1023,1024,1,12
%N A008949 Triangle of partial sums of binomial coefficients: T(n,k) =Sum_{i=0..k} 
               C(n,i); also dimensions of Reed-Muller codes.
%C A008949 The second-left-from-middle column is A000346: T(2n+2, n) = A000346(n). 
               - Ed Catmur (ed(AT)catmur.co.uk), Dec 09 2006
%C A008949 T(n,k) is the maximal number of regions into which n hyperplanes of co-dimension 
               1 divide R^k (the Cake-Without-Icing numbers) - Rob Johnson (rob(AT)whim.org), 
               Jul 27 2008
%D A008949 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting 
               Codes, Elsevier-North Holland, 1978, p. 376.
%H A008949 T. D. Noe, <a href="b008949.txt">Rows n=0..50 of triangle, flatten</a>
%H A008949 Rob Johnson, <a href="http://www.whim.org/nebula/math/spacediv.html">
               Dividing Space</a>.
%H A008949 <a href="Sindx_Pas.html#Pascal">Index entries for triangles and arrays 
               related to Pascal's triangle</a>
%F A008949 Form partial sums across rows of Pascal triangle A007318.
%F A008949 T(n, 0)=1, T(n, n)=2^n, T(n, k)=T(n-1, k-1)+T(n-1, k), 0<k<n.
%F A008949 G.f:(1 - x*y)/((1 - y - x*y)*(1 - 2*x*y)) [From Antonio Gonzalez (gonfer00(AT)gmail.com), 
               Sep 08 2009]
%F A008949 T(2n,n)=A032443(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Sep 16 2009]
%e A008949 1; 1,2; 1,3,4; 1,4,7,8; ...
%p A008949 A008949 := proc(n,k) local i; add(binomial(n,i),i=0..n)k end;
%t A008949 Table[Length[Select[Subsets[n], (Length[ # ] <= k) &]], {n, 0, 12}, {k, 
               0, n}] // Grid [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), 
               May 13 2009]
%Y A008949 Diagonals are given by A000079, A000225, A000295, A002663, A002664, A035038-A035042.
%Y A008949 T(n, m)= A055248(n, n-m).
%Y A008949 Cf. A110555, A007318.
%Y A008949 Cf. A000346.
%Y A008949 Sequence in context: A132110 A039912 A163311 this_sequence A076832 A078925 
               A072506
%Y A008949 Adjacent sequences: A008946 A008947 A008948 this_sequence A008950 A008951 
               A008952
%K A008949 tabl,nonn,easy,nice
%O A008949 0,3
%A A008949 N. J. A. Sloane (njas(AT)research.att.com).
%E A008949 More terms from Larry Reeves (larryr(AT)acm.org), Mar 23 2000