OFFSET
1,5
COMMENTS
Equivalently, the number of sets of rooted connected graphs on a total of n unlabeled nodes with a total of k blocks where every block is a complete graph.
Bivariate Euler transform of triangle A318602.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275
EXAMPLE
Triangle begins:
1;
1, 1;
1, 2, 2;
1, 3, 6, 4;
1, 4, 12, 16, 9;
1, 5, 20, 42, 46, 20;
1, 6, 30, 86, 145, 128, 48;
1, 7, 42, 153, 353, 483, 364, 115;
1, 8, 56, 248, 729, 1369, 1592, 1029, 286;
...
Case n=3: There are 5 sets of rooted graph which are illustrated below (an x marks a root node). These have 0, 1, 1, 2, 2 blocks so row 3 is 1, 2, 2.
x o o o o
/ / \ \ /
x x x x x---o x---o x---o
PROG
(PARI) \\ here EulerMT is Euler transform (bivariate version).
EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i ))-1)}
A(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerMT(y*EulerMT(v)))); [Vecrev(p) | p <- EulerMT(v)]}
{ my(T=A(10)); for(n=1, #T, print(T[n])) }
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Aug 30 2018
STATUS
approved