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.
g:= proc(n) option remember; `if`(n<2, n, add(g(n-k)*add(g(d)*d*
(-1)^(k/d+1), d=numtheory[divisors](k)), k=1..n-1)/(n-1))
end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(j-1-a(i), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> g(n)+b(n, n-1):
seq(a(n), n=1..40); # Alois P. Heinz, May 19 2022
g[n_] := g[n] = If[n < 2, n, Sum[g[n - k]*Sum[g[d]*d*(-1)^(k/d + 1), {d, Divisors[k]}], {k, 1, n - 1}]/(n - 1)];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[j - 1 - a[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := g[n] + b[n, n - 1];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, May 20 2022, after Alois P. Heinz *)
\\ here IdTreeGf is g.f. of A004111. IdTreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1) * d*A[d]) * A[n-k+1] ) ); x*Ser(A)} CHK(p,n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))} seq(n)={Vec(CHK(IdTreeGf(n), n))} \\ Andrew Howroyd, Aug 31 2018
For n = 3, the 7 endofunctions are (1,2,3) -> (1,1,1), (1,1,2), (1,2,1), (2,1,1), (1,2,3), (1,3,2) and (2,3,1). The components are respectively 123, 123, 13|2, 123, 1|2|3, 1|23 and 123; the number of components is thus 1, 1, 2, 1, 2, 3, 2, 1, so row 3 is 4,2,1. The triangle starts: 1; 2, 1; 4, 2, 1; 9, 7, 2, 1; 20, 17, 7, 2, 1;
Needs["Combinatorica`"];
nn=30;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2 k,0,s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i] s[n-1,i] i,{i,1,n-1}]/(n-1);rt=Table[a[i],{i,1,nn}];c=Drop[Apply[Plus,Table[Take[CoefficientList[CycleIndex[CyclicGroup[n],s]/.Table[s[j]->Table[Sum[rt[[i]] x^(k*i),{i,1,nn}],{k,1,nn}][[j]],{j,1,nn}],x],nn],{n,1,30}]],1];CoefficientList[Series[Product[1/(1-y x^i)^c[[i]],{i,1,nn-1}],{x,0,10}],{x,y}]//Grid (* Geoffrey Critzer, Oct 13 2012, after code given by Robert A. Russell in A000081 *)
a(0) = 0 because the null graph is trivially both partial functional and functional. a(1) = 1 because there are two partial functional graphs on one point: the point with, or without, a loop; the point with loop is the identity function, but without a loop the naked point is the unique merely partial functional case. a(2) = 3 because there are A126285(2) enumerates the 6 partial functional graphs on 2 points, of which 3 are functional, 6 - 3 = 3. a(3) = A126285(3) - A001372(3) = 16 - 7 = 9. a(4) = 45 - 19 = 26. a(5) = 121 - 47 = 74. a(6) = 338 - 130 = 208. a(7) = 929 - 343 = 586. a(8) = 2598 - 951 = 1647. a(9) = 7261 - 2615 = 4646. a(10) = 20453 - 7318 = 13135.
a(0) = 0, as the null digraph is formally neither connected nor disconnected. a(1) = 0, as the unique endofunction on one point is the identity function on one value and is connected. a(2) = 1, as there are 3 endofunctions on two points, two of which are "prime endofunctions" and one of which is the direct sum of two copies of the unique endofunction on one point, namely two points-with-loops, or the identity function on two values; 3 - 2 = 1. a(3) = A001372(3) - A002861(3) = 7 - 4 = 3. a(4) = A001372(4) - A002861(4) = 19 - 9 = 10. a(5) = A001372(5) - A002861(5) = 47 - 20 = 27. a(6) = 130 - 51 = 79. a(7) = 343 - 125 = 218. a(8) = 951 - 329 = 622. a(9) = 2615 - 862 = 1753. a(10) = 7318 - 2311 = 5007.
Triangle begins:
1;
3,
5, 2;
12, 7;
21, 25, 1;
58, 63, 9;
126, 178, 39;
341, 466, 140, 4;
867, 1253, 470, 25;
2334, 3418, 1431, 135;
6218, 9365, 4358, 544, 6;
17016, 25924, 12871, 2042, 50;
46351, 72207, 37993, 7056, 291;
127842, 202345, 111142, 23483, 1383, 4;
353297, 568822, 325359, 75701, 5754, 60;
T(3,2)=2 because (in the link) the third and the fifth digraphs on 3 nodes are composed of 2 unique components.
Needs["Combinatorica`"];
nn=30;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2 k,0,s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i] s[n-1,i] i,{i,1,n-1}]/(n-1);rt=Table[a[i],{i,1,nn}];c=Drop[Apply[Plus,Table[Take[CoefficientList[CycleIndex[CyclicGroup[n],s]/.Table[s[j]->Table[Sum[rt[[i]] x^(k*i),{i,1,nn}],{k,1,nn}][[j]],{j,1,nn}],x],nn],{n,1,30}]],1];CoefficientList[Series[Product[((y x^i +1-x^i)/(1-x^i))^c[[i]],{i,1,nn-1}],{x,0,15}],{x,y}]//Grid
(* after code given by Robert A. Russell in A000081 *)
Needs["Combinatorica`"];
nn=30;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2 k,0,s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i] s[n-1,i] i,{i,1,n-1}]/(n-1);rt=Table[a[i],{i,1,nn}];c=Drop[Apply[Plus,Table[Take[CoefficientList[CycleIndex[CyclicGroup[n],s]/.Table[s[j]->Table[Sum[rt[[i]] x^(k*i),{i,1,nn}],{k,1,nn}][[j]],{j,1,nn}],x],nn],{n,1,30}]],1];CoefficientList[Series[Product[(1+ x^i)^c[[i]],{i,1,nn-1}],{x,0,nn}],x] (* after code given by Robert A. Russell in A000081 *)
Triangle T(n,k) begins:
1;
1, 1;
3, 2, 1;
7, 6, 2, 1;
19, 16, 7, 2, 1;
47, 45, 19, 7, 2, 1;
130, 121, 57, 20, 7, 2, 1;
343, 338, 158, 60, 20, 7, 2, 1;
951, 929, 457, 170, 61, 20, 7, 2, 1;
...
nn = 10; A002861 = Cases[Import["https://oeis.org/A002861/b002861.txt", "Table"], {, }][[;; nn, 2]]; A000081 = Drop[Cases[ Import["https://oeis.org/A000081/b000081.txt", "Table"], {, }][[;; nn + 1, 2]], 1]; Map[Select[#, # > 0 &] &, CoefficientList[Series[ Product[1/(1 - y x^i)^A000081[[i]], {i, 1, nn}] Product[1/(1 - x^i)^A002861[[i]], {i, 1, nn}], {x, 0, nn}], {x,y}]] // Grid
Comments