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Triangular array read by rows: T(n,k) is the number of simple labeled graphs on n nodes whose maximal connected component has at most k nodes, n>=1, 1<=k<=n.
1

%I #13 Jul 26 2016 11:11:46

%S 1,1,2,1,4,8,1,10,26,64,1,26,106,296,1024,1,76,556,1696,6064,32768,1,

%T 232,2752,13392,43968,230896,2097152,1,764,15548,135248,461392,

%U 1956816,16886864,268435456,1,2620,99836,1062224,6932816,24877904,159248336,2423185664,68719476736,1,9496,636056,9621536,130702496,489604256,2281210016,24920583296,687883494016,35184372088832

%N Triangular array read by rows: T(n,k) is the number of simple labeled graphs on n nodes whose maximal connected component has at most k nodes, n>=1, 1<=k<=n.

%H Alois P. Heinz, <a href="/A275364/b275364.txt">Rows n = 1..82, flattened</a>

%e 1,

%e 1, 2,

%e 1, 4, 8,

%e 1, 10, 26, 64,

%e 1, 26, 106, 296, 1024,

%e 1, 76, 556, 1696, 6064, 32768,

%e 1, 232, 2752, 13392, 43968, 230896, 2097152,

%p with(combinat):

%p b:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-

%p add(k*binomial(n, k)*2^((n-k)*(n-k-1)/2)*b(k), k=1..n-1)/n)

%p end:

%p T:= proc(n, i) option remember; `if`(n=0, 1,

%p `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*

%p T(n-i*j, i-1)*b(i)^j, j=0..n/i)))

%p end:

%p seq(seq(T(n,k), k=1..n), n=1..12); # _Alois P. Heinz_, Jul 26 2016

%t nn = 10; f[z] := Sum[2^Binomial[n, 2] z^n/n!, {n, 0, nn}]; a = Drop[Range[0, nn]! CoefficientList[Series[Log[f[z]], {z, 0, nn}], z], 1]; Drop[Map[DeleteDuplicates,Transpose[Table[Range[0, nn]! CoefficientList[Series[Exp[Sum[a[[m]] z^m/m!, {m, 1, k}]], {z, 0, nn}], z], {k,1, nn}]]], 1] // Grid

%Y T(n,n) = A006125 for n>0.

%Y T(n,2) = A000085 for n>1.

%Y Cf. A001187.

%K nonn,tabl

%O 1,3

%A _Geoffrey Critzer_, Jul 24 2016