A361404 Triangle read by rows: T(n,k) is the number of graphs with loops on n unlabeled vertices with k loops.

1, 1, 1, 2, 2, 2, 4, 6, 6, 4, 11, 20, 28, 20, 11, 34, 90, 148, 148, 90, 34, 156, 544, 1144, 1408, 1144, 544, 156, 1044, 5096, 13128, 20364, 20364, 13128, 5096, 1044, 12346, 79264, 250240, 472128, 580656, 472128, 250240, 79264, 12346

Offset

0,4

Comments

T(n,k) is the number of bicolored graphs on n nodes with k vertices having the first color. Adjacent vertices may have the same color.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Formula

T(n,k) = T(n, n-k).

Example

Triangle begins:
     1;
     1,    1;
     2,    2,     2;
     4,    6,     6,     4;
    11,   20,    28,    20,    11;
    34,   90,   148,   148,    90,    34;
   156,  544,  1144,  1408,  1144,   544,  156;
  1044, 5096, 13128, 20364, 20364, 13128, 5096, 1044;
  ...

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) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
row(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)*prod(i=1, #p, 1 + x^p[i])); Vecrev(s/n!)}

Crossrefs

Columns k=0..2 are A000088, A000666(n-1), A303829.

Row sums are A000666.

Central coefficients are A361405.

Cf. A361361 (cubic).

Sequence in context: A342336 A320908 A356692 * A248781 A236840 A182539

Adjacent sequences: A361401 A361402 A361403 * A361405 A361406 A361407

Keyword

nonn,tabl

Author

Andrew Howroyd, Mar 11 2023

Status

approved