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, 4923897, 5703908, 6579783

Offset

1,2

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.

From Hugo Pfoertner, Feb 03 2026: (Start)

16*a(n) is also the sum of mutual Manhattan distances of the 2*n*(n+1)+1=A001844(n) distinct grid points (x_i,y_i) in a square lattice such that |x_i| + |y_i| <= n.

The coordinates of the grid points are provided in the first A001844(n) terms of A010751 and A305258.

16*a(n) >= A193452(A001844(n)):

n 16*a(n) m=A001844(n)

A193452(m)

1 16 5 16

2 192 13 188

3 1008 25 992

4 3504 41 3446

5 9504 61 9354

6 21840 85 21506

(End)

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seq., Vol. 7 (2004), Article 04.1.5.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

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

Sum_{n>=1} (-1)^(n+1)/a(n) = 480*Pi/17 - (140*Pi/17)*sech(sqrt(17/7)*Pi/2) - 250/3. - Amiram Eldar, Nov 02 2025

E.g.f.: exp(x)*x*(120 + 600*x + 600*x^2 + 175*x^3 + 14*x^4)/120. - Stefano Spezia, Dec 18 2025

Mathematica

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

Prog

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

Crossrefs

Cf. A006322, A006325, A079547, A085461, A085462, A085464, A085465.

Cf. A001844, A010751, A305258, A193452.

Sequence in context: A212509 A212249 A309372 * A051922 A027810 A391686

Adjacent sequences: A085460 A085461 A085462 * A085464 A085465 A085466

Keyword

nonn,easy

Author

Goran Kilibarda and Vladeta Jovovic, Jul 01 2003

Status

approved