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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.

%I A138107 #48 Jun 11 2023 12:26:10
%S A138107 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 A138107 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 A138107 2,11,52,291,1270,2423,1090,85,1,0,1,2,11,52,295,1596,5776,7388,2180,110,1,0
%N A138107 Infinite square array: T(n,k) = number of directed multigraphs with loops with n arcs and k vertices; read by falling antidiagonals.
%C A138107 Partial sums of the rows of A136564.
%H A138107 Andrew Howroyd, <a href="/A138107/b138107.txt">Table of n, a(n) for n = 0..1325</a>
%H A138107 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 A138107 T(n,k) = Sum_{p=0..k} A136564(n,p).
%F A138107 If k >= 2n, T(n,k) = A052171(n).
%e A138107 The array begins:
%e A138107    1, 1,   1,    1,     1,     1,     1,     1,     1, ...
%e A138107    0, 1,   2,    2,     2,     2,     2,     2,     2, ...
%e A138107    0, 1,   6,   10,    11,    11,    11,    11,    11, ...
%e A138107    0, 1,  10,   31,    47,    51,    52,    52,    52, ...
%e A138107    0, 1,  19,   90,   198,   269,   291,   295,   296,  296, ...
%e A138107    0, 1,  28,  222,   713,  1270,  1596,  1697,  1719, 1723, ...
%e A138107    0, 1,  44,  520,  2423,  5776,  8838, 10425, 10922, ...
%e A138107    0, 1,  60, 1090,  7388, 24032, 46384, ...
%e A138107    0, 1,  85, 2180, 21003, 93067, ...
%e A138107    0, 1, 110, 4090, ...
%e A138107    ...
%o A138107 (PARI)
%o A138107 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 A138107 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 A138107 G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)/edges(p,i->1-x^i)); s/n!}
%o A138107 T(n)={Mat(vector(n+1, k, Col(O(y*y^n) + G(k-1, y + O(y*y^n)))))}
%o A138107 {my(A=T(10)); for(n=1, #A, print(A[n,]))} \\ _Andrew Howroyd_, Oct 22 2019
%Y A138107 Columns k=0..4 are: A000007, A000012, A005993, A050927, A050929.
%Y A138107 Main diagonal is A362387.
%Y A138107 Cf. A052171, A136564, A333361.
%K A138107 nonn,tabl
%O A138107 0,8
%A A138107 _Benoit Jubin_, May 03 2008
%E A138107 More terms from _Vladeta Jovovic_ and _Benoit Jubin_, Sep 10 2008