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A228315 Triangular array read by rows: T(n,k) is the number of rooted labeled simple graphs on {1,2,...,n} such that the root is in a component of size k; n>=1, 1<=k<=n.

%I A228315 #29 Aug 27 2013 02:25:34
%S A228315 1,2,2,6,6,12,32,24,48,152,320,160,240,760,3640,6144,1920,1920,4560,
%T A228315 21840,160224,229376,43008,26880,42560,152880,1121568,13063792,
%U A228315 16777216,1835008,688128,680960,1630720,8972544,104510336,2012388736
%N A228315 Triangular array read by rows: T(n,k) is the number of rooted labeled simple graphs on {1,2,...,n} such that the root is in a component of size k; n>=1, 1<=k<=n.
%C A228315 Row sums = A095340.
%C A228315 Column 1 = A123903.
%C A228315 T(n,k)   = A223894(n,k)*k.
%C A228315 Diagonal = A053549.
%D A228315 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973, page 7.
%H A228315 Alois P. Heinz, <a href="/A228315/b228315.txt">Rows n = 1..45, flattened</a>
%H A228315 <a href="/index/Gra#graphs">Index entries for sequences related to graphs</a>
%F A228315 T(n,k) = binomial(n,k)*k*A001187(k)*A006125(n-k).
%e A228315 1;
%e A228315 2,    2;
%e A228315 6,    6,    12;
%e A228315 32,   24,   48,    152;
%e A228315 320,  160,  240,   760,    3640;
%p A228315 b:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
%p A228315       add(k*binomial(n, k)* 2^((n-k)*(n-k-1)/2)*b(k), k=1..n-1)/n)
%p A228315     end:
%p A228315 T:= (n, k)-> binomial(n, k)*k*b(k)*2^((n-k)*(n-k-1)/2):
%p A228315 seq(seq(T(n, k), k=1..n), n=1..10);  # _Alois P. Heinz_, Aug 26 2013
%t A228315 nn = 10; g = Sum[2^Binomial[n, 2] x^n/n!, {n, 0, nn}]; a =
%t A228315 Drop[Range[0, nn]! CoefficientList[Series[Log[g], {x, 0, nn}], x],
%t A228315   1]; Table[
%t A228315   Table[Binomial[n, k] k a[[k]] 2^Binomial[n - k, 2], {k, 1, n}], {n,
%t A228315    1, 7}] // Grid
%Y A228315 Cf. A070166.
%K A228315 nonn,tabl
%O A228315 1,2
%A A228315 _Geoffrey Critzer_, Aug 26 2013