A319729 Regular triangle read by rows where T(n,k) is the number of labeled simple graphs on n vertices where all non-isolated vertices have degree k.

1, 1, 1, 1, 3, 1, 1, 9, 7, 1, 1, 25, 37, 5, 1, 1, 75, 207, 85, 21, 1, 1, 231, 1347, 525, 591, 7, 1, 1, 763, 10125, 21385, 23551, 3535, 113, 1, 1, 2619, 86173, 180201, 1216701, 31647, 30997, 9, 1, 1, 9495, 819133, 12066705, 77636583, 66620631, 11485825, 286929, 955, 1

Offset

1,5

Links

Andrew Howroyd, Table of n, a(n) for n = 1..300

Formula

T(n,k) = Sum_{i=1..n} binomial(n,i)*A059441(i,k) for k > 0. - Andrew Howroyd, Dec 26 2020

Example

Triangle begins:
  1
  1       1
  1       3       1
  1       9       7       1
  1      25      37       5       1
  1      75     207      85      21       1
  1     231    1347     525     591       7       1
  1     763   10125   21385   23551    3535     113       1
  1    2619   86173  180201 1216701   31647   30997       9       1

Mathematica

Table[If[k==0, 1, Sum[Binomial[n, sup]*SeriesCoefficient[Product[1+Times@@x/@s, {s, Subsets[Range[sup], {2}]}], Sequence@@Table[{x[i], 0, k}, {i, sup}]], {sup, n}]], {n, 8}, {k, 0, n-1}]

Crossrefs

Row sums are A322555.

Cf. A000569, A005176, A058891, A059441, A295193, A301481, A306017, A306019, A306021, A319169, A319190, A319612.

Sequence in context: A294950 A294609 A204180 * A106340 A156610 A203460

Adjacent sequences: A319726 A319727 A319728 * A319730 A319731 A319732

Keyword

nonn,tabl

Author

Gus Wiseman, Dec 17 2018

Status

approved