A290776 - OEIS

%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