A333361 Array read by antidiagonals: T(n,k) is the number of directed loopless multigraphs with n arcs and k vertices.

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 2, 0, 0, 1, 1, 5, 2, 0, 0, 1, 1, 6, 10, 3, 0, 0, 1, 1, 6, 20, 24, 3, 0, 0, 1, 1, 6, 23, 69, 42, 4, 0, 0, 1, 1, 6, 24, 110, 196, 83, 4, 0, 0, 1, 1, 6, 24, 126, 427, 554, 132, 5, 0, 0, 1, 1, 6, 24, 129, 603, 1681, 1368, 222, 5, 0, 0

Offset

0,13

Links

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

Formula

T(n,k) = A052170(n) for k >= 2*n.

Example

==================================================
n\k | 0 1 2   3    4     5     6      7      8
----+---------------------------------------------
  0 | 1 1 1   1    1     1     1      1      1 ...
  1 | 0 0 1   1    1     1     1      1      1 ...
  2 | 0 0 2   5    6     6     6      6      6 ...
  3 | 0 0 2  10   20    23    24     24     24 ...
  4 | 0 0 3  24   69   110   126    129    130 ...
  5 | 0 0 3  42  196   427   603    668    684 ...
  6 | 0 0 4  83  554  1681  2983   3811   4116 ...
  7 | 0 0 4 132 1368  5881 13681  20935  24979 ...
  8 | 0 0 5 222 3240 19448 59680 112943 154504 ...
  ...

Prog

(PARI) \\ here G(k, x) gives column k as rational function.
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)^(2*g))) * prod(i=1, #v, t(v[i])^(v[i]-1))}
G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)/edges(p, i->1-x^i)); s/n!}
T(n)={Mat(vector(n+1, k, Col(O(y*y^n) + G(k-1, y + O(y*y^n)))))}
{my(A=T(10)); for(n=1, #A, print(A[n, ]))}

Crossrefs

Columns k=0..4 are A000007, A000007, A008619, A037240, A050930.

Cf. A052170, A138107, A192517, A290428.

Sequence in context: A190164 A253938 A131488 * A363665 A308183 A346632

Adjacent sequences: A333358 A333359 A333360 * A333362 A333363 A333364

Keyword

nonn,tabl

Author

Andrew Howroyd, Mar 16 2020

Status

approved