A275364 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, 1, 2, 1, 4, 8, 1, 10, 26, 64, 1, 26, 106, 296, 1024, 1, 76, 556, 1696, 6064, 32768, 1, 232, 2752, 13392, 43968, 230896, 2097152, 1, 764, 15548, 135248, 461392, 1956816, 16886864, 268435456, 1, 2620, 99836, 1062224, 6932816, 24877904, 159248336, 2423185664, 68719476736, 1, 9496, 636056, 9621536, 130702496, 489604256, 2281210016, 24920583296, 687883494016, 35184372088832

Offset

1,3

Links

Alois P. Heinz, Rows n = 1..82, flattened

Example

1,
1, 2,
1, 4,   8,
1, 10,  26,   64,
1, 26,  106,  296,   1024,
1, 76,  556,  1696,  6064,  32768,
1, 232, 2752, 13392, 43968, 230896, 2097152,

Maple

with(combinat):
b:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
      add(k*binomial(n, k)*2^((n-k)*(n-k-1)/2)*b(k), k=1..n-1)/n)
    end:
T:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
       T(n-i*j, i-1)*b(i)^j, j=0..n/i)))
    end:
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Jul 26 2016

Mathematica

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

Crossrefs

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

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

Cf. A001187.

Sequence in context: A221632 A071951 A264059 * A160323 A340469 A128411

Adjacent sequences: A275361 A275362 A275363 * A275365 A275366 A275367

Keyword

nonn,tabl

Author

Geoffrey Critzer, Jul 24 2016

Status

approved