A370319 - OEIS

%I #11 Feb 28 2024 06:30:18

%S 1,1,0,1,0,1,1,0,3,4,1,0,6,16,41,1,0,10,40,205,768,1,0,15,80,615,4608,

%T 27449,1,0,21,140,1435,16128,192143,1887284,1,0,28,224,2870,43008,

%U 768572,15098272,252522481,1,0,36,336,5166,96768,2305716,67942224,2272702329,66376424160

%N Triangle read by rows where T(n,k) is the number of labeled graphs with n vertices and k non-isolated vertices.

%F T(n,k) = binomial(n,k) * A006129(k).

%F T(n,n-1) = (n-1) * A006129(n-1).

%F T(n,k) = A198261(n, n-k). - _Andrew Howroyd_, Feb 26 2024

%e Triangle begins:

%e      1

%e      1     0

%e      1     0     1

%e      1     0     3     4

%e      1     0     6    16    41

%e      1     0    10    40   205   768

%e      1     0    15    80   615  4608 27449

%e Row n = 3 counts the following edge sets:

%e   {}  .  {{1,2}}  {{1,2},{1,3}}

%e          {{1,3}}  {{1,2},{2,3}}

%e          {{2,3}}  {{1,3},{2,3}}

%e                   {{1,2},{1,3},{2,3}}

%t Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Length[Union@@#]==k&]],{n,0,5},{k,0,n}]

%t Flatten@Table[Binomial[n,k]*Sum[(-1)^(k-m) Binomial[k,m] 2^Binomial[m,2],{m,0,k}],{n,0,10},{k,0,n}] (* _Giorgos Kalogeropoulos_, Feb 25 2024 *)

%Y Row sums are A006125, unlabeled A000088.

%Y Column k = n is A006129, unlabeled A002494.

%Y Mirror of A198261, unlabeled A217653.

%Y The unlabeled version is the partial subsequences of A002494.

%Y Cf. A001187, A003465, A006126, A116508, A143543, A287689, A367862.

%K nonn,tabl

%O 0,9

%A _Gus Wiseman_, Feb 18 2024

%E More terms from _Giorgos Kalogeropoulos_, Feb 25 2024