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A063258 a(n) = binomial(n+5,4) - 1. 11

%I

%S 4,14,34,69,125,209,329,494,714,1000,1364,1819,2379,3059,3875,4844,

%T 5984,7314,8854,10625,12649,14949,17549,20474,23750,27404,31464,35959,

%U 40919,46375,52359,58904,66044,73814,82250,91389,101269,111929,123409,135750

%N a(n) = binomial(n+5,4) - 1.

%C In the Frey-Sellers reference this sequence is called {(n+2) over 4}_{3}, n >= 0.

%C If X is an n-set and Y a fixed (n-4)-subset of X then a(n-5) is equal to the number of 4-subsets of X intersecting Y. - _Milan Janjic_, Aug 15 2007

%C For n>=5, a(n-5) is the number of permutations of 1,2...,n with the distribution of up (1) - down (0) elements 0...01000 (the first n-5 zeros), or, the same, a(n-5) is up-down coefficient {n,8} (see comment in A060351). - _Vladimir Shevelev_, Feb 18 2014

%H Harry J. Smith, <a href="/A063258/b063258.txt">Table of n, a(n) for n = 0..1000</a>

%H Guillaume Aupy, Julien Herrmann. <a href="http://dx.doi.org/10.1080%2F10556788.2016.1230612">Periodicity in optimal hierarchical checkpointing schemes for adjoint computations</a>. Optimization Methods and Software, Volume 32, 2017 - Issue 3. <a href="http://people.bordeaux.inria.fr/gaupy/ressources/pub/journals/oms_periodicity.pdf">Preprint</a>

%H D. D. Frey and J. A. Sellers, <a href="http://www.fq.math.ca/Scanned/39-2/frey.pdf">Generalizing Bailey's generalization of the Catalan numbers</a>, The Fibonacci Quarterly, 39 (2001) 142-148.

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = A062750(n+2, 4) = (n+6)*(n+1)*(n^2 + 7*n + 16)/4!.

%F G.f.: (x-2)*(x^2-2*x+2)/(x-1)^5 = N(4;1, x)/(1-x)^5 with N(4;1, x)= 4 - 6*x + 4*x^2 - x^3, polynomial of second row of A062751.

%p [seq(binomial(n+5,4)-1,n=0..37)]; # _Zerinvary Lajos_, Nov 25 2006

%t s1=s2=s3=s4=0;lst={};Do[a=n+(n+2);s1+=a;s2+=s1;s3+=s2;s4+=s3;If[(s3-2)/2>0,AppendTo[lst,(s3-2)/2]],{n,-1,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Apr 04 2009 *)

%o (PARI) { for (n=0, 1000, write("b063258.txt", n, " ", binomial(n + 5, 4) - 1) ) } \\ _Harry J. Smith_, Aug 19 2009

%Y Fifth column (r=4) of FS(4) staircase array A062750.

%Y A column of triangle A014473.

%Y Cf. A000096, A062748.

%K nonn,easy

%O 0,1

%A _Wolfdieter Lang_, Jul 12 2001

%E Simpler definition from _Vladeta Jovovic_, Jul 21 2003

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Last modified May 27 21:12 EDT 2020. Contains 334671 sequences. (Running on oeis4.)