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.
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 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