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A361404 Triangle read by rows: T(n,k) is the number of graphs with loops on n unlabeled vertices with k loops.

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