A085463 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.

1, 12, 63, 219, 594, 1365, 2786, 5202, 9063, 14938, 23529, 35685, 52416, 74907, 104532, 142868, 191709, 253080, 329251, 422751, 536382, 673233, 836694, 1030470, 1258595, 1525446, 1835757, 2194633, 2607564, 3080439, 3619560, 4231656

Offset: 1

Author

Goran Kilibarda, Vladeta Jovovic, Jul 01 2003

Keywords

• nonn
• easy

Comments

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A006322, A006325, A079547, A085461-A085465.

Programs

Magma

[n*(n+1)*(2*n+1)*(7*n^2 + 7*n + 6)/120: n in [1..25]]; // G. C. Greubel, Oct 07 2017

Mathematica

Table[n*(n+1)*(2*n+1)*(7*n^2 + 7*n + 6)/120, {n,1,25}] (* G. C. Greubel, Oct 07 2017 *)

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

Formula

a(n) = n + 10*binomial(n, 2) + 30*binomial(n, 3) + 35*binomial(n, 4) + 14*binomial(n, 5).

a(n) = n*(n+1)*(2*n+1)*(7*n^2 + 7*n + 6)/120.

G.f.: x*(1+6*x+6*x^2+x^3)/(1-x)^6. - Colin Barker, Apr 01 2012