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%I #23 Dec 17 2022 12:42:24
%S 1,19,134,586,1919,5173,12124,25572,49677,90343,155650,256334,406315,
%T 623273,929272,1351432,1922649,2682363,3677374,4962706,6602519,
%U 8671069,11253716,14447980,18364645,23128911,28881594,35780374,44001091
%N Number of monotone n-weightings of complete bipartite digraph K(4,2).
%C 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="/A085464/b085464.txt">Table of n, a(n) for n = 1..1000</a>
%H Goran Kilibarda and Vladeta Jovovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Kilibarda/kili2.html">Antichains of Multisets</a>, J. Integer Seqs., Vol. 7, 2004.
%F a(n) = n + 17*binomial(n, 2) + 80*binomial(n, 3) + 160*binomial(n, 4) + 144*binomial(n, 5) + 48*binomial(n, 6).
%F a(n) = (1/30)*n*(n+1)*(2*n^4+4*n^3+6*n^2+4*n-1).
%F a(n) = Sum_{i=1..n} ((n+1-i)^4-(n-i)^4)*i^2.
%F a(n) = Sum_{i=1..n} ((n+1-i)^2-(n-i)^2)*i^4.
%F 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 G.f.: x*(1+x)^2*(1+10*x+x^2)/(1-x)^7. - _Colin Barker_, Apr 01 2012
%F a(n) = sum(i=1..n, sum (j=1..n, min(i,j)^4)). - _Enrique Pérez Herrero_, Jan 16 2013
%t Table[(1/30)*n*(n+1)*(2*n^4+4*n^3+6*n^2+4*n-1), {n,1,50}] (* _G. C. Greubel_, Oct 07 2017 *)
%o (PARI) a(n)=n*(n+1)*(2*n^4+4*n^3+6*n^2+4*n-1)/30 \\ _Charles R Greathouse IV_, Jan 16 2013
%o (Magma) [(1/30)*n*(n+1)*(2*n^4+4*n^3+6*n^2+4*n-1): n in [1..25]]; // _G. C. Greubel_, Oct 07 2017
%Y Cf. A006322, A006325, A079547, A085461, A085462, A085463, A085465.
%K nonn,easy
%O 1,2
%A Goran Kilibarda and _Vladeta Jovovic_, Jul 01 2003
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