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%I #29 Jan 10 2024 00:29:16
%S 1,14,77,273,748,1729,3542,6630,11571,19096,30107,45695,67158,96019,
%T 134044,183260,245973,324786,422617,542717,688688,864501,1074514,
%U 1323490,1616615,1959516,2358279,2819467,3350138,3957863,4650744
%N Number of 5-tuples (v1,v2,v3,v4,v5) of nonnegative integers less than n such that v1<=v4, v1<=v5, v2<=v4 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.
%C Dimensions of certain Lie algebra (see Landsberg-Manivel reference for precise definition). - _N. J. A. Sloane_, Oct 15 2007
%H G. C. Greubel, <a href="/A085462/b085462.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.
%H J. M. Landsberg and L. Manivel, <a href="https://doi.org/10.1016/j.aim.2005.02.001">The sextonions and E7 1/2</a>, Adv. Math. 201 (2006), 143-179. [Th. 7.1, case a=-2/3]
%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 + 12*binomial(n, 2) + 38*binomial(n, 3) + 45*binomial(n, 4) + 18*binomial(n, 5).
%F a(n) = n*(n+1)*(2*n+1)*(3*n+1)*(3*n+2)/120.
%F G.f.: x*(1+x)*(1+7*x+x^2)/(1-x)^6. - _Colin Barker_, Apr 01 2012
%F a(n) = sum(i=1..n, sum(j=1..n, i^2 * Min(i,j))). - _Enrique Pérez Herrero_, Jan 30 2013
%t Rest[CoefficientList[Series[x*(1 + x)*(1 + 7*x + x^2)/(1 - x)^6, {x, 0, 50}], x]] (* or *) Table[n*(n+1)*(2*n+1)*(3*n+1)*(3*n+2)/120, {n,0,50}] (* _G. C. Greubel_, Oct 07 2017 *)
%o (PARI) x='x+O('x^50); Vec(x*(1+x)*(1+7*x+x^2)/(1-x)^6) \\ _G. C. Greubel_, Oct 07 2017
%o (Magma) [n*(n+1)*(2*n+1)*(3*n+1)*(3*n+2)/120: n in [0..50]]; // _G. C. Greubel_, Oct 07 2017
%Y Cf. A006322, A006325, A079547, A085461, A085463, A085464, A085465.
%K nonn,easy
%O 1,2
%A Goran Kilibarda and _Vladeta Jovovic_, Jul 01 2003
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