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 A286030 #37 Oct 10 2018 10:02:37
%S A286030 1,1,2,1,6,2,1,8,6,12,1,10,20,20,30,1,12,30,30,20,120,30,1,14,42,42,
%T A286030 70,210,140,210,1,16,56,56,112,336,70,560,420,560,1,18,72,72,168,504,
%U A286030 252,1008,756,630,2520,560
%N A286030 Irregular triangle T(n,k) read by rows: Let S be a 3-member set of integers {f,g,h} where f >= g >= h >= 0 and f+g+h = n. Let S(n,k) be an irregular triangle composed of all S listed in reverse lexicographic order by row n. Then T(n,k) = n!*Q/(3*f!*g!*h!), where Q is the number of permutations of S(n,k). (See "Comments" and "Examples" for additional explanation.)
%C A286030 See "Example" below for the starting construction of S(n,k) and T(n,k).
%C A286030 To understand S(n,k) and Q, consider example S(5,k), i.e., f+g+h = 5, and S(n,k) are listed in reverse lexicographic order. So S(5,k) = {5,0,0}, {4,1,0}, {3,2,0}, {3,1,1}, {2,2,1} k=1..5, respectively. Q is the number of permutations of S(n,k). So Q=3 when S(n,k) = {5,0,0}, {3,1,1} and {2,2,1}; and Q=6 when S(n,k) = {4,1,0} and {3,2,0}.
%C A286030 In general, by definition: Q=1 when all members of S(n,k) are equal, Q=3 when S(n,k) contains a pair, and Q=6 when none of the members of S(n,k) is equal.
%C A286030 Suppose three equally-matched players are playing a tournament of n games; and for each game there is one winner and two losers. Then S(n,k) is the "overall win record" (where player order does not matter) after n games. Let p be the probability that any S(n,k) occurs after n games. Then p = T(n,k)/3^(n-1). (See also "Example" section.)
%C A286030 Generally, when S(n,k) is a z-member set {f,g,h,i..,y}, then Q is the number of permutations of S(n,k), T(n,k) = n!*Q/(z*f!*g!*h!..*y!) and p = T(n,k)/z^(n-1). So when z=2 we get A008314. (Observation prompted by query from Linda Rogers.)
%C A286030 For triangle T(n,k):
%C A286030 Row sums are 3^(n-1).
%C A286030 Row lengths are A001399(n).
%C A286030 Final terms in each row are A199127(n).
%C A286030 For n >= 3: T(n,2) = 2*n.
%C A286030 For n >= 5: T(n,3) = T(n,4) = A002378(n-1) (oblong numbers).
%C A286030 For n >= 6: T(n,6) = A007531(n).
%C A286030 For n >= 8: T(n,9) = A033487(n-3).
%H A286030 Nicolas Behr, Pawel Sobocinski, <a href="https://arxiv.org/abs/1807.00785">Rule Algebras for Adhesive Categories</a>, arXiv:1807.00785 [cs.LO], 2018, also <a href="https://doi.org/10.4230/LIPIcs.CSL.2018.11">LIPIcs 27th EACSL Annual Conference on Computer Science Logic</a> (CSL 2018), Vol. 119, pp. 11:1-11:21.
%e A286030 Triangle T(n,k) begins:
%e A286030 n/k 1 2 3 4 5 6 7 8 9 10 11 12 13 14
%e A286030 1: 1
%e A286030 2: 1, 2
%e A286030 3: 1, 6, 2
%e A286030 4: 1, 8, 6, 12
%e A286030 5: 1, 10, 20, 20, 30
%e A286030 6: 1, 12, 30, 30, 20, 120, 30
%e A286030 7: 1, 14, 42, 42, 70, 210, 140, 210
%e A286030 8: 1, 16, 56, 56, 112, 336, 70, 560, 420, 560
%e A286030 9: 1, 18, 72, 72, 168, 504, 252, 1008, 756, 630, 2520, 560
%e A286030 10: 1, 20, 90, 90, 240, 720, 420, 1680, 1260, 252, 2520, 5040, 3150, 4200
%e A286030 Triangle S(n,k) begins:
%e A286030 n/k 1 2 3 4 5 6 7
%e A286030 1: {1,0,0}
%e A286030 2: {2,0,0} {1,1,0}
%e A286030 3: {3,0,0} {2,1,0} {1,1,1}
%e A286030 4: {4,0,0} {3,1,0} {2,2,0} {2,1,1}
%e A286030 5: {5,0,0} {4,1,0} {3,2,0} {3,1,1} {2,2,1}
%e A286030 6: {6,0,0} {5,1,0} {4,2,0} {4,1,1} {3,3,0} {3,2,1} {2,2,2}
%e A286030 T(4,3) = 6 because n=4 and S(4,3) = {2,2,0}; so Q=3 and 3*4!/(3*2!*2!*0!) = 6. Therefore p = 6/27 = 2/9 that the overall win record = {2,2,0} after playing 4 tournament games.
%Y A286030 Cf. A001399, A002378, A007531, A008314, A033487, A199127.
%Y A286030 Cf. A000041 (partition numbers).
%K A286030 nonn,tabf
%O A286030 1,3
%A A286030 _Bob Selcoe_, Apr 30 2017