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%I #13 Mar 11 2023 19:48:24
%S 1,1,1,2,2,2,4,6,6,4,11,20,28,20,11,34,90,148,148,90,34,156,544,1144,
%T 1408,1144,544,156,1044,5096,13128,20364,20364,13128,5096,1044,12346,
%U 79264,250240,472128,580656,472128,250240,79264,12346
%N Triangle read by rows: T(n,k) is the number of graphs with loops on n unlabeled vertices with k loops.
%C T(n,k) is the number of bicolored graphs on n nodes with k vertices having the first color. Adjacent vertices may have the same color.
%H Andrew Howroyd, <a href="/A361404/b361404.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%F T(n,k) = T(n, n-k).
%e Triangle begins:
%e 1;
%e 1, 1;
%e 2, 2, 2;
%e 4, 6, 6, 4;
%e 11, 20, 28, 20, 11;
%e 34, 90, 148, 148, 90, 34;
%e 156, 544, 1144, 1408, 1144, 544, 156;
%e 1044, 5096, 13128, 20364, 20364, 13128, 5096, 1044;
%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) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
%o row(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)*prod(i=1, #p, 1 + x^p[i])); Vecrev(s/n!)}
%Y Columns k=0..2 are A000088, A000666(n-1), A303829.
%Y Row sums are A000666.
%Y Central coefficients are A361405.
%Y Cf. A361361 (cubic).
%K nonn,tabl
%O 0,4
%A _Andrew Howroyd_, Mar 11 2023
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