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Infinite square array: T(n,k) = number of directed multigraphs with loops with n arcs and k vertices; read by falling antidiagonals.
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%I #48 Jun 11 2023 12:26:10

%S 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,

%T 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,

%U 2,11,52,291,1270,2423,1090,85,1,0,1,2,11,52,295,1596,5776,7388,2180,110,1,0

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

%C Partial sums of the rows of A136564.

%H Andrew Howroyd, <a href="/A138107/b138107.txt">Table of n, a(n) for n = 0..1325</a>

%H R. J. Mathar, <a href="http://arxiv.org/abs/1709.09000">Statistics on Small Graphs</a>, arXiv:1709.09000 [math.CO] (2017) Table 79.

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

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

%e The array begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 0, 1, 2, 2, 2, 2, 2, 2, 2, ...

%e 0, 1, 6, 10, 11, 11, 11, 11, 11, ...

%e 0, 1, 10, 31, 47, 51, 52, 52, 52, ...

%e 0, 1, 19, 90, 198, 269, 291, 295, 296, 296, ...

%e 0, 1, 28, 222, 713, 1270, 1596, 1697, 1719, 1723, ...

%e 0, 1, 44, 520, 2423, 5776, 8838, 10425, 10922, ...

%e 0, 1, 60, 1090, 7388, 24032, 46384, ...

%e 0, 1, 85, 2180, 21003, 93067, ...

%e 0, 1, 110, 4090, ...

%e ...

%o (PARI)

%o 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}

%o 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])}

%o G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)/edges(p,i->1-x^i)); s/n!}

%o T(n)={Mat(vector(n+1, k, Col(O(y*y^n) + G(k-1, y + O(y*y^n)))))}

%o {my(A=T(10)); for(n=1, #A, print(A[n,]))} \\ _Andrew Howroyd_, Oct 22 2019

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

%Y Main diagonal is A362387.

%Y Cf. A052171, A136564, A333361.

%K nonn,tabl

%O 0,8

%A _Benoit Jubin_, May 03 2008

%E More terms from _Vladeta Jovovic_ and _Benoit Jubin_, Sep 10 2008