A138107 Infinite square array: T(n,k) = number of directed multigraphs with loops with n arcs and k vertices; read by falling antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 6, 1, 0, 1, 2, 10, 10, 1, 0, 1, 2, 11, 31, 19, 1, 0, 1, 2, 11, 47, 90, 28, 1, 0, 1, 2, 11, 51, 198, 222, 44, 1, 0, 1, 2, 11, 52, 269, 713, 520, 60, 1, 0, 1, 2, 11, 52, 291, 1270, 2423, 1090, 85, 1, 0, 1, 2, 11, 52, 295, 1596, 5776, 7388, 2180, 110, 1, 0

Offset: 0

Author

Benoit Jubin, May 03 2008

Keywords

• nonn
• tabl

Comments

Partial sums of the rows of A136564.

Examples

			The array begins:
   1, 1,   1,    1,     1,     1,     1,     1,     1, ...
   0, 1,   2,    2,     2,     2,     2,     2,     2, ...
   0, 1,   6,   10,    11,    11,    11,    11,    11, ...
   0, 1,  10,   31,    47,    51,    52,    52,    52, ...
   0, 1,  19,   90,   198,   269,   291,   295,   296,  296, ...
   0, 1,  28,  222,   713,  1270,  1596,  1697,  1719, 1723, ...
   0, 1,  44,  520,  2423,  5776,  8838, 10425, 10922, ...
   0, 1,  60, 1090,  7388, 24032, 46384, ...
   0, 1,  85, 2180, 21003, 93067, ...
   0, 1, 110, 4090, ...
   ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325
• R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017) Table 79.

Crossrefs

Columns k=0..4 are: A000007, A000012, A005993, A050927, A050929.

Main diagonal is A362387.

Cf. A052171, A136564, A333361.

Programs

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(v[i]*v[j]/g)^(2*g))) * prod(i=1, #v, t(v[i])^v[i])}
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,]))} \\ Andrew Howroyd, Oct 22 2019

Formula

T(n,k) = Sum_{p=0..k} A136564(n,p).

If k >= 2n, T(n,k) = A052171(n).

Extensions

More terms from Vladeta Jovovic and Benoit Jubin, Sep 10 2008