OFFSET
0,8
COMMENTS
Are the row sums the same as column k = 1 (shifted left)?
Yes. When k = 1 there is one loop. Remove the vertex with the loop and add loops to its neighbors. This process is reversible so there is a bijection. - Andrew Howroyd, Jan 13 2024
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
EXAMPLE
Triangle begins:
1
0 1
0 1 1
1 2 2 1
2 6 6 2 1
6 17 18 8 2 1
21 52 58 30 9 2 1
Representatives of the loop-graphs counted by row n = 4:
{12}{13}{14}{23} {1}{12}{13}{14} {1}{2}{12}{13} {1}{2}{3}{12} {1}{2}{3}{4}
{12}{13}{24}{34} {1}{12}{13}{23} {1}{2}{12}{34} {1}{2}{3}{14}
{1}{12}{13}{24} {1}{2}{13}{14}
{1}{12}{23}{24} {1}{2}{13}{23}
{1}{12}{23}{34} {1}{2}{13}{24}
{1}{23}{24}{34} {1}{2}{13}{34}
MATHEMATICA
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]], p[[i]]}, {i, Length[p]}])], {p, Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n], {1, 2}], {n}], Count[#, {_}]==k&]]], {n, 0, 4}, {k, 0, n}]
PROG
(PARI)
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}
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)))}
row(n) = {my(s=0, A=1+O(x*x^n)); forpart(p=n, s+=permcount(p) * polcoef(edges(p, i->A + x^i)*prod(i=1, #p, A + (x*y)^p[i]), n)); Vecrev(s/n!)} \\ Andrew Howroyd, Jan 13 2024
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Jan 11 2024
EXTENSIONS
a(28) onwards from Andrew Howroyd, Jan 13 2024
STATUS
approved