This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A353996 #6 May 13 2022 10:45:18 %S A353996 1,4,36,752,45960,9133760,6154473664,14334221970688, %T A353996 117222686206799936,3412369204476033220608, %U A353996 357745172369222114451432448,136400229481294592916607770361856,190697841181900458854914389940360337408 %N A353996 Rooted simple digraphs on n unlabeled vertices. %C A353996 Loops and parallel edges are not permitted. %C A353996 There are four ways that a vertex other than the root can be joined to the root: only towards, only away, both, neither. Remove the root and color the remaining vertices by four colors corresponding to how they were joined to the root. This gives a bijection with 4-colored digraphs on n-1 vertices, which is A329874(n-1,4). %F A353996 a(n) = A329874(n-1,4). %p A353996 with(Iterator): %p A353996 RootedDig := proc(n) %p A353996 local i,j,ptn,ans,a,orb2,orb4,hasptn,nextptn; %p A353996 (hasptn,nextptn) := ModuleIterator(PartitionPartCount(n-1)); %p A353996 ans := 0; %p A353996 while hasptn() do %p A353996 ptn := nextptn(); %p A353996 a := 1 / mul(j^ptn[j]*ptn[j]!,j=1..n-1); %p A353996 orb2 := add(ptn[j],j=2..n-1,2); %p A353996 orb4 := add(ptn[j]*j/2,j=2..n-1,2) %p A353996 + add(ptn[j]*(j+1)/2,j=1..n-1,2) %p A353996 + add(ptn[j]*(ptn[j]-1)*j/2,j=1..n-1) %p A353996 + add(add(ptn[i]*ptn[j]*igcd(i,j),i=1..j-1),j=2..n-1); %p A353996 ans := ans + a*2^orb2*4^orb4; %p A353996 end do; %p A353996 ans; %p A353996 end proc; %Y A353996 Cf. A329874. %K A353996 nonn %O A353996 1,2 %A A353996 _Brendan McKay_, May 13 2022