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A138107
Infinite square array: T(n,k) = number of directed multigraphs with loops with n arcs and k vertices; read by falling antidiagonals.
14
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
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.
Sequence in context: A138352 A129620 A074766 * A089631 A332032 A298878
KEYWORD
nonn,tabl
AUTHOR
Benoit Jubin, May 03 2008
EXTENSIONS
More terms from Vladeta Jovovic and Benoit Jubin, Sep 10 2008
STATUS
approved