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%I #23 Aug 13 2018 09:09:05
%S 1,0,1,0,1,3,0,1,7,16,0,1,12,63,125,0,1,18,162,722,1296,0,1,25,341,
%T 2565,10140,16807,0,1,33,636,7180,47100,169137,262144,0,1,42,1092,
%U 17335,168285,987567,3271576,4782969,0,1,52,1764,37750,509545,4364017,23315936,72043092,100000000
%N Triangle T(n,k) read by rows: the number of connected, loopless, non-oriented, vertex-labeled graphs with n >= 0 edges and k >= 1 vertices, allowing multi-edges.
%C This is the vertex-labeled companion to A191646.
%H Andrew Howroyd, <a href="/A290776/b290776.txt">Table of n, a(n) for n = 0..1274</a>
%H R. J. Mathar, <a href="http://arxiv.org/abs/1709.09000">Statistics on Small Graphs</a>, arXiv:1709.09000 [math.CO] (2017), Table 62.
%e The triangle starts in row n=0 with 1 <= k <= n+1 vertices as
%e 1;
%e 0, 1;
%e 0, 1, 3;
%e 0, 1, 7, 16;
%e 0, 1, 12, 63, 125;
%e 0, 1, 18, 162, 722, 1296;
%e 0, 1, 25, 341, 2565, 10140, 16807;
%e 0, 1, 33, 636, 7180, 47100, 169137, 262144;
%e 0, 1, 42, 1092, 17355, 168285, 987567, 3271576, 4782969;
%e 0, 1, 52, 1764, 37750, 509545, 4364017, 23315936, 72043092, 100000000;
%e ...
%t S[m_, n_] := Binomial[Binomial[m, 2] + n - 1, n];
%t R[nn_] := Module[{cc = Array[0&, {nn, nn}]}, cc[[1, 1]] = 1; For[m = 1, m <= nn, m++, For[n = 1, n <= nn-1, n++, cc[[m, n+1]] = S[m, n] - S[m-1, n] - Sum[Sum[Binomial[m-1, i-1]*cc[[i, j+1]]*S[m-i, n-j], {j, 1, n}], {i, 2, m-1}]]]; cc // Transpose];
%t A = R[10];
%t Table[A[[n, k]], {n, 1, Length[A]}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Aug 13 2018, after _Andrew Howroyd_ *)
%o (PARI) \\ here S(m,n) is m nodes with n edges, not necessarily connected
%o S(m,n)={ binomial(binomial(m,2) + n - 1, n) }
%o R(N)={ my(C=matrix(N,N)); C[1,1]=1; for(m=1, N, for(n=1, N-1, C[m,n+1] = S(m,n) - S(m-1,n) - sum(i=2, m-1, sum(j=1, n, binomial(m-1, i-1)*C[i,j+1]*S(m-i, n-j))))); C~; }
%o { my(A=R(10)); for(n=1, #A, for(k=1, n, print1(A[n,k],", ")); print) } \\ _Andrew Howroyd_, May 13 2018
%Y Cf. A055998 (k=3), A000272 (diagonal), A060091 (k=4?), A060093 (k=5?).
%K nonn,tabl
%O 0,6
%A _R. J. Mathar_, Aug 10 2017
%E Terms a(34) and beyond from _Andrew Howroyd_, May 13 2018
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