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A085464 Number of monotone n-weightings of complete bipartite digraph K(4,2). 3
1, 19, 134, 586, 1919, 5173, 12124, 25572, 49677, 90343, 155650, 256334, 406315, 623273, 929272, 1351432, 1922649, 2682363, 3677374, 4962706, 6602519, 8671069, 11253716, 14447980, 18364645, 23128911, 28881594, 35780374, 44001091 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

FORMULA

a(n) = n + 17*binomial(n, 2) + 80*binomial(n, 3) + 160*binomial(n, 4) + 144*binomial(n, 5) + 48*binomial(n, 6).

a(n) = (1/30)*n*(n+1)*(2*n^4+4*n^3+6*n^2+4*n-1).

a(n) = Sum_{i=1..n} ((n+1-i)^4-(n-i)^4)*i^2.

a(n) = Sum_{i=1..n} ((n+1-i)^2-(n-i)^2)*i^4.

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.

G.f.: x*(1+x)^2*(1+10*x+x^2)/(1-x)^7. - Colin Barker, Apr 01 2012

a(n) = sum(i=1..n, sum (j=1..n, min(i,j)^4)). - Enrique Pérez Herrero, Jan 16 2013

MATHEMATICA

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 *)

PROG

(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

(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

CROSSREFS

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

Sequence in context: A108673 A041692 A078977 * A215863 A101090 A213122

Adjacent sequences:  A085461 A085462 A085463 * A085465 A085466 A085467

KEYWORD

nonn,easy

AUTHOR

Goran Kilibarda, Vladeta Jovovic, Jul 01 2003

STATUS

approved

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Last modified September 21 11:18 EDT 2019. Contains 327253 sequences. (Running on oeis4.)