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A123551 Triangle read by rows: T(n,k) gives number of unlabeled graphs without endpoints on n nodes and k edges, (n >= 0, 0 <= k <= n(n-1)/2).

%I A123551 #13 Feb 07 2024 20:26:47
%S A123551 1,1,1,0,1,0,0,1,1,0,0,1,1,1,1,1,0,0,1,1,2,4,3,2,1,1,1,0,0,1,1,2,6,8,
%T A123551 13,16,13,8,5,2,1,1,1,0,0,1,1,2,6,10,22,48,76,97,102,84,60,39,20,10,5,
%U A123551 2,1,1,1,0,0,1,1,2,6,10,25,64,152,331,617,930,1173,1253,1140
%N A123551 Triangle read by rows: T(n,k) gives number of unlabeled graphs without endpoints on n nodes and k edges, (n >= 0, 0 <= k <= n(n-1)/2).
%D A123551 R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.
%H A123551 R. W. Robinson, <a href="/A123551/b123551.txt">Rows 0 through 16, flattened</a>
%F A123551 T(n,k) = A008406(n,k) - A240168(n,k). - _Andrew Howroyd_, Apr 16 2021
%e A123551 Triangle begins:
%e A123551 [0] 1;
%e A123551 [1] 1;
%e A123551 [2] 1, 0;
%e A123551 [3] 1, 0, 0, 1;
%e A123551 [4] 1, 0, 0, 1, 1, 1, 1;
%e A123551 [5] 1, 0, 0, 1, 1, 2, 4, 3,  2,  1,  1;
%e A123551 [6] 1, 0, 0, 1, 1, 2, 6, 8, 13, 16, 13, 8, 5, 2, 1, 1;
%e A123551   ...
%o A123551 (PARI)
%o A123551 permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
%o A123551 edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
%o A123551 row(n) = {my(s=0); sum(k=0, n, forpart(p=k, s+=permcount(p) * edges(p, w->1+y^w) * y^(n-k)*polcoef(prod(i=1, #p, 1-x^p[i]), n-k)/k!)); Vecrev(s, binomial(n,2)+1)}
%o A123551 { for(n=0, 6, print(row(n))) } \\ _Andrew Howroyd_, Feb 07 2024
%Y A123551 Row sums are A004110.
%Y A123551 Cf. A008406, A240168, A369928 (labeled).
%K A123551 nonn,tabf
%O A123551 0,21
%A A123551 _N. J. A. Sloane_, Nov 14 2006