OFFSET
1,3
COMMENTS
These graphs correspond to "trivially ungraceful" graphs that do not have enough integers less than or equal to the edge count to cover all the vertices.
LINKS
Eric Weisstein's World of Mathematics, Graceful Graph
Eric Weisstein's World of Mathematics, Simple Graph
Eric Weisstein's World of Mathematics, Ungraceful Graph
FORMULA
a(n) <= A308556(n).
a(n) = Sum_{k=0..n-2} A008406(n, k). - Andrew Howroyd, Apr 08 2020
MATHEMATICA
Get["Combinatorica`"] // Quiet;
Table[Total[Take[CoefficientList[GraphPolynomial[n, x], x], n - 1]], {n, 20}]
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)))}
a(n)={my(s=0); if(n>1, forpart(p=n, s+=permcount(p)*polcoef(edges(p, i->1 + x^i + O(x^(n-1)))/(1-x), n-2) )); s/n!} \\ Andrew Howroyd, Apr 08 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Apr 08 2020
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Apr 08 2020
STATUS
approved