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%I #19 Sep 08 2022 08:45:11
%S 1,12,63,219,594,1365,2786,5202,9063,14938,23529,35685,52416,74907,
%T 104532,142868,191709,253080,329251,422751,536382,673233,836694,
%U 1030470,1258595,1525446,1835757,2194633,2607564,3080439,3619560,4231656
%N Number of 5-tuples (v1,v2,v3,v4,v5) of nonnegative integers less than n such that v1<=v4, v1<=v5, v2<=v4, v2<=v5 and v3<=v4.
%C Number of monotone n-weightings of a certain connected bipartite digraph. 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.
%H G. C. Greubel, <a href="/A085463/b085463.txt">Table of n, a(n) for n = 1..1000</a>
%H 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 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F a(n) = n + 10*binomial(n, 2) + 30*binomial(n, 3) + 35*binomial(n, 4) + 14*binomial(n, 5).
%F a(n) = n*(n+1)*(2*n+1)*(7*n^2 + 7*n + 6)/120.
%F G.f.: x*(1+6*x+6*x^2+x^3)/(1-x)^6. - _Colin Barker_, Apr 01 2012
%t Table[n*(n+1)*(2*n+1)*(7*n^2 + 7*n + 6)/120, {n,1,25}] (* _G. C. Greubel_, Oct 07 2017 *)
%o (PARI) for(n=1,25, print1(n*(n+1)*(2*n+1)*(7*n^2 + 7*n + 6)/120, ", ")) \\ _G. C. Greubel_, Oct 07 2017
%o (Magma) [n*(n+1)*(2*n+1)*(7*n^2 + 7*n + 6)/120: n in [1..25]]; // _G. C. Greubel_, Oct 07 2017
%Y Cf. A006322, A006325, A079547, A085461-A085465.
%K nonn,easy
%O 1,2
%A Goran Kilibarda, _Vladeta Jovovic_, Jul 01 2003
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