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 A348634 #41 Mar 19 2023 18:58:01
%S A348634 0,0,0,27,768,8771,63468,340620,1470784,5371002,17153352,49075521,
%T A348634 128066400,309124101,697874996,1486830618,3011414784,5833686340,
%U A348634 10863883728,19532496375,34028554944,57623258007,95101946940,153331834040,241997811264,374544148830,569365964440,851301035325,1253479866912,1819599953913,2606698902276
%N A348634 Number of transitive relations on an n-set with exactly five ordered pairs.
%F A348634 a(n) = 27*C(n,3) + 660*C(n,4) + 5201*C(n,5) + 21822*C(n,6) + 54600*C(n,7) + 84000*C(n,8) + 75600*C(n,9) + 30240*C(n,10).
%F A348634 a(n) = (1/120)*(n^10 - 20*n^9 + 220*n^8 - 1500*n^7 + 6710*n^6 - 19954*n^5 + 38765*n^4 - 46950*n^3 + 31944*n^2 - 9216*n).
%F A348634 a(n) = C(n,3)*(n^7 - 17*n^6 + 167*n^5 - 965*n^4 + 3481*n^3 - 7581*n^2 + 9060*n - 4608)/20. - _Chai Wah Wu_, Jan 06 2022
%e A348634 No relation containing exactly five ordered pairs on a 2-element set exists. Thus a(2)=0.
%e A348634 Also, there are 27 transitive relations with exactly five ordered pairs on a 3-set. One such relation is {(1,1),(1,2),(1,3),(2,2),(3,2)} on the 3-set {1,2,3}.
%o A348634 (Python)
%o A348634 def A348634(n): return n*(n - 2)*(n - 1)*(n*(n*(n*(n*(n*(n*(n - 17) + 167) - 965) + 3481) - 7581) + 9060) - 4608)//120 # _Chai Wah Wu_, Jan 06 2022
%Y A348634 Cf. A349919, A349927, A349849, A006905.
%K A348634 nonn
%O A348634 0,4
%A A348634 _Firdous Ahmad Mala_, Dec 13 2021
%E A348634 a(9) corrected by _Georg Fischer_, Mar 19 2023