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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 (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 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. Cf. A000664, A007718, A007719, A054923, A191646, A275421, A321254, A322114, A322115, A322137. Sequence in context: A171616 A323883 A008290 * A059066 A059067 A356707 Adjacent sequences: A322144 A322145 A322146 * A322148 A322149 A322150 KEYWORD nonn,tabl AUTHOR Gus Wiseman, Nov 28 2018 EXTENSIONS Terms a(28) and beyond from Andrew Howroyd, Nov 29 2018 STATUS approved

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Last modified September 15 11:45 EDT 2024. Contains 375938 sequences. (Running on oeis4.)