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A277219 Triangle read by rows: T(n,k) is the number of independent sets of size k over all simple labeled graphs on n nodes, n>=0, 0<=k<=n.

%I A277219 #19 Oct 07 2016 07:57:24
%S A277219 1,1,1,2,4,1,8,24,12,1,64,256,192,32,1,1024,5120,5120,1280,80,1,32768,
%T A277219 196608,245760,81920,7680,192,1,2097152,14680064,22020096,9175040,
%U A277219 1146880,43008,448,1,268435456,2147483648,3758096384,1879048192,293601280,14680064,229376,1024,1
%N A277219 Triangle read by rows: T(n,k) is the number of independent sets of size k over all simple labeled graphs on n nodes, n>=0, 0<=k<=n.
%C A277219 Equivalently, T(n,k) is the number of size k cliques over all simple labeled graphs on n vertices.
%H A277219 Robert Israel, <a href="/A277219/b277219.txt">Table of n, a(n) for n = 0..3402</a> (rows 0 to 81, flattened)
%F A277219 T(n,k) = 2^binomial(n,2)*binomial(n,k)/2^binomial(k,2).
%e A277219 Triangle begins:
%e A277219 1;
%e A277219 1,     1;
%e A277219 2,     4,      1;
%e A277219 8,     24,     12,     1;
%e A277219 64,    256,    192,    32,    1;
%e A277219 1024,  5120,   5120,   1280,  80,   1;
%e A277219 32768, 196608, 245760, 81920, 7680, 192, 1;
%e A277219 ...
%p A277219 seq(seq(2^(n*(n-1)/2-k*(k-1)/2)*binomial(n,k),k=0..n),n=0..10); # _Robert Israel_, Oct 06 2016
%t A277219 Table[Table[2^Binomial[n, 2] Binomial[n, k]/2^Binomial[k, 2], {k, 0, n}], {n,0, 7}] // Grid
%Y A277219 Cf. A079491 (row sums), A006125 (column k=0), A095340 (column k=1), A095351 (column k = 2).
%K A277219 nonn,tabl
%O A277219 0,4
%A A277219 _Geoffrey Critzer_, Oct 05 2016