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 A349919 #59 Aug 26 2025 15:53:39
%S A349919 0,0,5,27,90,230,495,945,1652,2700,4185,6215,8910,12402,16835,22365,
%T A349919 29160,37400,47277,58995,72770,88830,107415,128777,153180,180900,
%U A349919 212225,247455,286902,330890,379755,433845,493520,559152,631125,709835,795690,889110,990527,1100385,1219140,1347260,1485225,1633527,1792670
%N A349919 Number of transitive relations on an n-set with exactly two ordered pairs.
%H A349919 Firdous Ahmad Mala, <a href="http://dx.doi.org/10.26855/jamc.2021.12.002">Counting Transitive Relations with Two Ordered Pairs</a>, Journal of Applied Mathematics and Computation, 5(4), 247-251.
%H A349919 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A349919 a(n) = 5*C(n,2) + 12*C(n,3) + 12*C(n,4).
%F A349919 a(n) = (1/2)*(n^4 - 2*n^3 + 4*n^2 - 3*n).
%F A349919 a(n) = A336535(n) - 1.
%F A349919 From _Elmo R. Oliveira_, Aug 26 2025: (Start)
%F A349919 G.f.: x^2*(5 + 2*x + 5*x^2)/(1 - x)^5.
%F A349919 E.g.f.: x^2*(5 + 4*x + x^2)*exp(x)/2.
%F A349919 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)
%e A349919 a(2) = 5. The five relations on a 2-set are {(1,1),(1,2)}, {(1,1),(2,1)}, {(1,1),(2,2)}, {(1,2),(2,2)} and {(2,1),(2,2)}.
%t A349919 LinearRecurrence[{5,-10,10,-5,1},{0,0,5,27,90},50] (* _Harvey P. Dale_, Oct 23 2022 *)
%Y A349919 Cf. A006905, A336535, A349927.
%Y A349919 This is a diagonal of the array A285192.
%K A349919 nonn,easy,changed
%O A349919 0,3
%A A349919 _Firdous Ahmad Mala_, Dec 05 2021