A085461 Number of 5-tuples (v1,v2,v3,v4,v5) of nonnegative integers less than n such that v1 <= v5, v2 <= v5, v2 <= v4 and v3 <= v4.

1, 13, 70, 246, 671, 1547, 3164, 5916, 10317, 17017, 26818, 40690, 59787, 85463, 119288, 163064, 218841, 288933, 375934, 482734, 612535, 768867, 955604, 1176980, 1437605, 1742481, 2097018, 2507050, 2978851, 3519151, 4135152, 4834544

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.

Kekulé numbers for certain benzenoids. - Emeric Deutsch, Nov 18 2005

Can be constructed by taking the product of the three members of a Pythagorean triples and dividing by 60. Formula: n*(n^2-1)*(n^2+1)/240 where n runs through the odd numbers >= 3. - Pierre Gayet, Apr 04 2009

Number of composable morphisms in a height-n tower of retractions. A retraction between objects X and Y is a pair of maps s:X->Y and r:Y->X such that r(s(x))=x for all x in X. Given objects X_0,X_1,X_2,...,X_n, we can ask for retractions s_i:X_i->X_{i+1},r_i:X_{i+1}->X_i, for each 0 <= i < n. The total number of morphisms in that category is 0^2 + 1^2 + 2^2 + ... + n^2 (cf. A000330). The total number of composable pairs of morphisms in that category is the sequence given here. - David Spivak, Feb 26 2014

References

• S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 168).

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.
• Daeseok Lee and H.-K. Ju, An Extension of Hibi's palindromic theorem, arXiv preprint arXiv:1503.05658 [math.CO], 2015.
• R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
• See p. 31
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A006322, A006325, A079547, A085462, A085463, A085464, A085465.

Programs

Mathematica

Rest[CoefficientList[Series[x*(1 + x)*(1 + 6*x + x^2)/(1 - x)^6, {x, 0, 50}], x]] (* G. C. Greubel, Oct 06 2017 *)

PARI

x='x+O('x^50); Vec(x*(1+x)*(1+6*x+x^2)/(1-x)^6) \\ G. C. Greubel, Oct 06 2017

Formula

a(n) = n + 11*binomial(n, 2) + 34*binomial(n, 3) + 40*binomial(n, 4) + 16*binomial(n, 5) = 1/30*n*(n+1)*(2*n+1)*(2*n^2 + 2*n + 1).

From Bruno Berselli, Dec 27 2010: (Start)

G.f.: x*(1+x)*(1+6*x+x^2)/(1-x)^6.

a(n) = ( n*A110450(n) - Sum_{i=0..n-1} A110450(i) )/3. (End)