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 A085465 #32 Sep 08 2022 08:45:11
%S A085465 1,15,102,442,1443,3885,9100,19188,37269,67771,116754,192270,304759,
%T A085465 467481,696984,1013608,1442025,2011815,2758078,3722082,4951947,
%U A085465 6503365,8440356,10836060,13773565,17346771,21661290,26835382,33000927,40304433
%N A085465 Number of monotone n-weightings of complete bipartite digraph K(3,3).
%C A085465 A monotone n-(vertex) weighting of a digraph D=(V,E) is a function w: V -> {0,1,..,n-1} such that w(v1)<=w(v2) for every arc (v1,v2) from E.
%C A085465 a(n) = number of proper mergings of a 3-antichain and an (n-1)-chain. - _Henri Mühle_, Aug 17 2012
%H A085465 G. C. Greubel, <a href="/A085465/b085465.txt">Table of n, a(n) for n = 1..1000</a>
%H A085465 Goran Kilibarda and Vladeta Jovovic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Kilibarda/kili2.html">Antichains of Multisets</a>, J. Integer Seqs., Vol. 7, 2004.
%H A085465 H. Mühle, <a href="http://arxiv.org/abs/1206.3922">Counting Proper Mergings of Chains and Antichains</a>, arXiv:1206.3922 [math.CO], 2012.
%H A085465 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A085465 a(n) = n + 13*binomial(n, 2) + 60*binomial(n, 3) + 120*binomial(n, 4) + 108*binomial(n, 5) + 36*binomial(n, 6) = 1/20*n*(n+1)*(n^2+1)*(n^2+2*n+2) = Sum_{i=1..n} ((n+1-i)^3-(n-i)^3)*i^3. More generally, number of monotone n-weightings of complete bipartite digraph K(s, t) is Sum_{i=1..n} ((n+1-i)^s-(n-i)^s)*i^t = Sum_{i=1..n} ((n+1-i)^t-(n-i)^t)*i^s.
%F A085465 G.f.: x*(1+4*x+x^2)^2/(1-x)^7. - _Colin Barker_, Apr 01 2012
%F A085465 a(n) = A006003(n)*A006003(n+1)/5 for n>0. - _Bruno Berselli_, Jun 26 2018
%t A085465 Rest[CoefficientList[Series[x*(1 + 4*x + x^2)^2/(1 - x)^7, {x, 0, 50}], x]] (* _G. C. Greubel_, Oct 06 2017 *)
%t A085465 LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {1, 15, 102, 442, 1443, 3885, 9100}, 40] (* _Vincenzo Librandi_, Oct 06 2017 *)
%o A085465 (PARI) x='x+O('x^50); Vec(x*(1+4*x+x^2)^2/(1-x)^7) \\ _G. C. Greubel_, Oct 06 2017
%o A085465 (Magma) [1/20*n*(n+1)*(n^2+1)*(n^2+2*n+2): n in [1..40]]; // _Vincenzo Librandi_, Oct 06 2017
%Y A085465 Cf. A006003, A006322, A006325, A079547, A085461-A085465.
%K A085465 nonn,easy
%O A085465 1,2
%A A085465 Goran Kilibarda, _Vladeta Jovovic_, Jul 01 2003