A333361 Array read by antidiagonals: T(n,k) is the number of directed loopless multigraphs with n arcs and k vertices.
1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 2, 0, 0, 1, 1, 5, 2, 0, 0, 1, 1, 6, 10, 3, 0, 0, 1, 1, 6, 20, 24, 3, 0, 0, 1, 1, 6, 23, 69, 42, 4, 0, 0, 1, 1, 6, 24, 110, 196, 83, 4, 0, 0, 1, 1, 6, 24, 126, 427, 554, 132, 5, 0, 0, 1, 1, 6, 24, 129, 603, 1681, 1368, 222, 5, 0, 0
Offset: 0
Examples
================================================== n\k | 0 1 2 3 4 5 6 7 8 ----+--------------------------------------------- 0 | 1 1 1 1 1 1 1 1 1 ... 1 | 0 0 1 1 1 1 1 1 1 ... 2 | 0 0 2 5 6 6 6 6 6 ... 3 | 0 0 2 10 20 23 24 24 24 ... 4 | 0 0 3 24 69 110 126 129 130 ... 5 | 0 0 3 42 196 427 603 668 684 ... 6 | 0 0 4 83 554 1681 2983 3811 4116 ... 7 | 0 0 4 132 1368 5881 13681 20935 24979 ... 8 | 0 0 5 222 3240 19448 59680 112943 154504 ... ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1325 (antidiagonals 0..50)
Crossrefs
Programs
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PARI
\\ here G(k,x) gives column k as rational function. 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]-1))} 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, ]))}
Formula
T(n,k) = A052170(n) for k >= 2*n.