%I #50 Feb 11 2020 02:05:01
%S 1,13,70,246,671,1547,3164,5916,10317,17017,26818,40690,59787,85463,
%T 119288,163064,218841,288933,375934,482734,612535,768867,955604,
%U 1176980,1437605,1742481,2097018,2507050,2978851,3519151,4135152,4834544
%N 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.
%C 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.
%C Kekulé numbers for certain benzenoids. - _Emeric Deutsch_, Nov 18 2005
%C 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
%C 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
%D S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 168).
%H G. C. Greubel, <a href="/A085461/b085461.txt">Table of n, a(n) for n = 1..1000</a>
%H Goran Kilibarda and Vladeta Jovovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Kilibarda/kili2.html">Antichains of Multisets</a>, J. Integer Seqs., Vol. 7, 2004.
%H Daeseok Lee and H.-K. Ju, <a href="http://arxiv.org/abs/1503.05658">An Extension of Hibi's palindromic theorem</a>, arXiv preprint arXiv:1503.05658 [math.CO], 2015.
%H R. P. Stanley, <a href="/A002721/a002721.pdf">Examples of Magic Labelings</a>, Unpublished Notes, 1973 [Cached copy, with permission]
%H See p. 31
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F 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).
%F From _Bruno Berselli_, Dec 27 2010: (Start)
%F G.f.: x*(1+x)*(1+6*x+x^2)/(1-x)^6.
%F a(n) = ( n*A110450(n) - Sum_{i=0..n-1} A110450(i) )/3. (End)
%t Rest[CoefficientList[Series[x*(1 + x)*(1 + 6*x + x^2)/(1 - x)^6, {x, 0, 50}], x]] (* _G. C. Greubel_, Oct 06 2017 *)
%o (PARI) x='x+O('x^50); Vec(x*(1+x)*(1+6*x+x^2)/(1-x)^6) \\ _G. C. Greubel_, Oct 06 2017
%Y Cf. A006322, A006325, A079547, A085462, A085463, A085464, A085465.
%K nonn,easy
%O 1,2
%A Goran Kilibarda, _Vladeta Jovovic_, Jul 01 2003