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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). 5

%I #13 Feb 07 2024 20:26:47

%S 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 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 2,1,1,1,0,0,1,1,2,6,10,25,64,152,331,617,930,1173,1253,1140

%N 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 R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.

%H R. W. Robinson, <a href="/A123551/b123551.txt">Rows 0 through 16, flattened</a>

%F T(n,k) = A008406(n,k) - A240168(n,k). - _Andrew Howroyd_, Apr 16 2021

%e Triangle begins:

%e [0] 1;

%e [1] 1;

%e [2] 1, 0;

%e [3] 1, 0, 0, 1;

%e [4] 1, 0, 0, 1, 1, 1, 1;

%e [5] 1, 0, 0, 1, 1, 2, 4, 3, 2, 1, 1;

%e [6] 1, 0, 0, 1, 1, 2, 6, 8, 13, 16, 13, 8, 5, 2, 1, 1;

%e ...

%o (PARI)

%o 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 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 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 { for(n=0, 6, print(row(n))) } \\ _Andrew Howroyd_, Feb 07 2024

%Y Row sums are A004110.

%Y Cf. A008406, A240168, A369928 (labeled).

%K nonn,tabf

%O 0,21

%A _N. J. A. Sloane_, Nov 14 2006

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Last modified March 28 11:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)