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A370063 Triangle read by rows: T(n,k) is the number of unlabeled loopless multigraphs without endpoints with n edges covering k vertices, 0 <= k <= n.

%I A370063 #10 Feb 09 2024 01:41:54
%S A370063 1,0,0,0,0,1,0,0,1,1,0,0,1,2,2,0,0,1,3,5,2,0,0,1,5,14,10,4,0,0,1,6,25,
%T A370063 33,18,4,0,0,1,8,46,96,90,31,7,0,0,1,10,75,227,330,194,52,8,0,0,1,12,
%U A370063 117,494,1033,962,416,82,12,0,0,1,14,173,982,2847,3908,2591,800,128,14
%N A370063 Triangle read by rows: T(n,k) is the number of unlabeled loopless multigraphs without endpoints with n edges covering k vertices, 0 <= k <= n.
%C A370063 An endpoint is a vertex that appears in only one edge. Equivalently, the degree of every vertex >= 2.
%H A370063 Andrew Howroyd, <a href="/A370063/b370063.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%e A370063 Triangle begins:
%e A370063   1;
%e A370063   0, 0;
%e A370063   0, 0, 1;
%e A370063   0, 0, 1,  1;
%e A370063   0, 0, 1,  2,   2;
%e A370063   0, 0, 1,  3,   5,   2;
%e A370063   0, 0, 1,  5,  14,  10,    4;
%e A370063   0, 0, 1,  6,  25,  33,   18,   4;
%e A370063   0, 0, 1,  8,  46,  96,   90,  31,   7;
%e A370063   0, 0, 1, 10,  75, 227,  330, 194,  52,  8;
%e A370063   0, 0, 1, 12, 117, 494, 1033, 962, 416, 82, 12;
%e A370063   ...
%o A370063 (PARI)
%o A370063 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 A370063 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 A370063 G(n) = {my(s=O(x*x^n)); sum(k=0, n, forpart(p=k, s+=permcount(p) / edges(p, w->1-y^w+O(y*y^n)) * x^k * prod(i=1, #p, 1-(y*x)^p[i], 1+O(x^(n-k+1))) / k!)); s*(1-x)}
%o A370063 T(n)={my(r=Vec(substvec(G(n), [x, y], [y, x]))); vector(#r, i, Vecrev(Pol(r[i]), i)) }
%o A370063 { my(A=T(10)); for(i=1, #A, print(A[i])) }
%Y A370063 Row sums are A307316.
%Y A370063 Main diagonal is A002865.
%Y A370063 Cf. A369932.
%K A370063 nonn,tabl
%O A370063 0,14
%A A370063 _Andrew Howroyd_, Feb 08 2024