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

0,8

Comments

Partial sums of the rows of A136564.

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.

Formula

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

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

Example

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, ...
   ...

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(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

Crossrefs

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

Main diagonal is A362387.

Cf. A052171, A136564, A333361.

Sequence in context: A138352 A129620 A074766 * A089631 A332032 A298878

Adjacent sequences: A138104 A138105 A138106 * A138108 A138109 A138110

Keyword

nonn,tabl

Author

Benoit Jubin, May 03 2008

Extensions

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

Status

approved