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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. 4
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.

Cf. A000664, A007718, A007719, A054923, A191646, A275421, A321254, A322114, A322115, A322137.

Sequence in context: A171616 A323883 A008290 * A059066 A059067 A065861

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 May 30 08:04 EDT 2020. Contains 334712 sequences. (Running on oeis4.)