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A322148 Regular triangle where T(n,k) is the number of labeled connected multigraphs with loops with n edges and k vertices. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified July 13 13:48 EDT 2020. Contains 335688 sequences. (Running on oeis4.)