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

1,2

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.

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R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]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).

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)

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 *)

Prog

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

Crossrefs

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

Sequence in context: A105058 A146469 A146381 * A081860 A050403 A235454

Adjacent sequences: A085458 A085459 A085460 * A085462 A085463 A085464

Keyword

nonn,easy

Author

Goran Kilibarda, Vladeta Jovovic, Jul 01 2003

Status

approved