A123548 Triangle read by rows: T(n,k) = number of unlabeled bicolored graphs having 2n nodes and k edges, which are invariant when the two color classes are interchanged. Here n >= 0, 0 <= k <= n^2.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 2, 1, 1, 1, 1, 1, 1, 2, 4, 5, 7, 8, 9, 8, 7, 5, 4, 2, 1, 1, 1, 1, 1, 1, 2, 4, 6, 9, 14, 22, 29, 33, 37, 43, 43, 37, 33, 29, 22, 14, 9, 6, 4, 2, 1, 1, 1, 1, 1, 1, 2, 4, 6, 10, 16, 29, 46, 69, 99, 141, 183, 230, 277, 319, 342, 352, 342, 319, 277, 230, 183, 141, 99, 69, 46, 29, 16, 10, 6, 4, 2, 1, 1, 1

Offset

0,12

References

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1978.

Links

Andrew Howroyd, Rows n=0..20, flattened (rows 0..7 from R. W. Robinson)

Example

Triangle begins:
n = 0
k = 0 : 1
************************ total ( 2n = 0) = 1
n = 1
k = 0 : 1
k = 1 : 1
************************ total ( 2n = 2) = 2
n = 2
k = 0 : 1
k = 1 : 1
k = 2 : 1
k = 3 : 1
k = 4 : 1
************************ total ( 2n = 4) = 5
n = 3
k = 0 : 1
k = 1 : 1
k = 2 : 1
k = 3 : 2
k = 4 : 3
k = 5 : 3
k = 6 : 2
k = 7 : 1
k = 8 : 1
k = 9 : 1
************************ total ( 2n = 6) = 16

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(2*v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(2*c)^(c\2)*if(c%2, t(c), 1))}
Row(n) = {my(s=0); forpart(p=n, s+=permcount(p)*edges(p, i->1+x^i)); Vecrev(s/n!)}
{ for(n=0, 6, print(Row(n))) } \\ Andrew Howroyd, Mar 08 2020

Crossrefs

Row sums give A122082.

Cf. A008406.

Sequence in context: A074989 A307429 A261283 * A131838 A274885 A287732

Adjacent sequences: A123545 A123546 A123547 * A123549 A123550 A123551

Keyword

nonn,tabf

Author

N. J. A. Sloane, Nov 14 2006

Status

approved