A322148 Regular triangle where T(n,k) is the number of labeled connected multigraphs with loops with n edges and k vertices.

1, 1, 1, 1, 3, 3, 1, 6, 16, 16, 1, 10, 51, 127, 125, 1, 15, 126, 574, 1347, 1296, 1, 21, 266, 1939, 8050, 17916, 16807, 1, 28, 504, 5440, 35210, 135156, 286786, 262144, 1, 36, 882, 13387, 125730, 736401, 2642122, 5368728, 4782969, 1, 45, 1452, 29854, 388190, 3239491, 17424610, 58925728, 115089813, 100000000

Offset

0,5

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Example

Triangle begins:
  1
  1     1
  1     3     3
  1     6    16    16
  1    10    51   127   125
  1    15   126   574  1347  1296
  1    21   266  1939  8050 17916 16807

Mathematica

multsubs[set_, k_]:=If[k==0, {{}}, Join@@Table[Prepend[#, set[[i]]]&/@multsubs[Drop[set, i-1], k-1], {i, Length[set]}]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Union[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[If[n==0, 1, Length[Select[multsubs[multsubs[Range[k], 2], n], And[Union@@#==Range[k], Length[csm[#]]==1]&]]], {n, 0, 5}, {k, 1, n+1}]

Prog

(PARI)
Connected(v)={my(u=vector(#v)); for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1, k)*v[k]*u[n-k])); u}
M(n)={Mat([Col(p, -(n+1)) | p<-Connected(vector(2*n, j, 1/(1 - x + O(x*x^n) )^binomial(j+1, 2)))[1..n+1]])}
{ my(T=M(10)); for(n=1, #T, print(T[n, ][1..n])) } \\ Andrew Howroyd, Nov 29 2018

Crossrefs

Row sums are A322152. Last column is A000272.

Cf. A007718, A191646, A191970, A275421, A321155, A322114, A322115, A322137, A322147.

Sequence in context: A117279 A234251 A049323 * A084144 A306759 A214362

Adjacent sequences: A322145 A322146 A322147 * A322149 A322150 A322151

Keyword

nonn,tabl

Author

Gus Wiseman, Nov 28 2018

Extensions

Offset corrected and terms a(28) and beyond from Andrew Howroyd, Nov 29 2018

Status

approved