OFFSET
0,18
COMMENTS
All such graphs are cactus graphs (with bridges allowed).
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1750 (rows 0..100)
Wikipedia, Cactus graph.
FORMULA
T(3*n, n) = A380634(n).
EXAMPLE
Triangle begins:
1;
0;
0;
0, 1;
0, 1;
0, 1;
0, 1, 1;
0, 1, 1;
0, 1, 2;
0, 1, 2, 2;
0, 1, 3, 5;
0, 1, 3, 10;
0, 1, 4, 17, 6;
0, 1, 4, 26, 18;
0, 1, 5, 38, 51;
0, 1, 5, 52, 106, 18;
...
PROG
(PARI)
EulerMTS(p)={my(n=serprec(p, x)-1, vars=variables(p)); exp(sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i))}
raise(p, d) = {my(n=serprec(p, x)-1); substvec(p + O(x^(n\d+1)), [x, y], [x^d, y^d])}
R(n, y)={my(g = O(x^3)); for(n=1, (n-1)\2, my(p=x*EulerMTS(g), p2=raise(p, 2)); g=p*y*(p^2/(1 - p) + (1 + p)*p2/(1 - p2))/2); g}
G(n, y=1)={my(g=R(n, y), p = x*EulerMTS(g) + O(x*x^n));
my( r=((1 + p)^2/(1 - raise(p, 2)) - 1)/2 );
my( c=-sum(d=1, n, eulerphi(d)/d*log(raise(1-p, d))) );
1 + (raise(g, 2) - g^2 + y*(r + c - 2*p - p^2 - raise(p, 2)))/2 }
T(n)={[Vecrev(p) | p<-Vec(G(n, y))]}
{ my(A=T(15)); for(i=1, #A, print(A[i])) }
CROSSREFS
KEYWORD
nonn,tabf,new
AUTHOR
Andrew Howroyd, Feb 24 2025
STATUS
approved