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A322147
Regular triangle read by rows where T(n,k) is the number of labeled connected graphs with loops with n edges and k vertices, 1 <= k <= n+1.
6
1, 1, 1, 0, 2, 3, 0, 1, 10, 16, 0, 0, 12, 79, 125, 0, 0, 6, 162, 847, 1296, 0, 0, 1, 179, 2565, 11436, 16807, 0, 0, 0, 116, 4615, 47100, 185944, 262144, 0, 0, 0, 45, 5540, 121185, 987567, 3533720, 4782969, 0, 0, 0, 10, 4720, 220075, 3376450, 23315936, 76826061, 100000000
OFFSET
0,5
LINKS
EXAMPLE
Triangle begins:
1
1 1
0 2 3
0 1 10 16
0 0 12 79 125
0 0 6 162 847 1296
0 0 1 179 2565 11436 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[Subsets[multsubs[Range[k], 2], {n}], And[Union@@#==Range[k], Length[csm[#]]==1]&]]], {n, 0, 6}, {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 + 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 A322151. Last column is A000272.
Column sums are A062740.
Sequence in context: A171616 A323883 A008290 * A059066 A059067 A356707
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Nov 28 2018
EXTENSIONS
Terms a(28) and beyond from Andrew Howroyd, Nov 29 2018
STATUS
approved