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

%I A063258 #42 Jan 05 2025 19:51:36
%S A063258 4,14,34,69,125,209,329,494,714,1000,1364,1819,2379,3059,3875,4844,
%T A063258 5984,7314,8854,10625,12649,14949,17549,20474,23750,27404,31464,35959,
%U A063258 40919,46375,52359,58904,66044,73814,82250,91389,101269,111929,123409,135750
%N A063258 a(n) = binomial(n+5,4) - 1.
%C A063258 In the Frey-Sellers reference this sequence is called {(n+2) over 4}_{3}, n >= 0.
%C A063258 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 A063258 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 A063258 Harry J. Smith, <a href="/A063258/b063258.txt">Table of n, a(n) for n = 0..1000</a>
%H A063258 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 A063258 D. D. Frey and J. A. Sellers, <a href="https://web.archive.org/web/2024*/https://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 A063258 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H A063258 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A063258 a(n) = A062750(n+2, 4) = (n+6)*(n+1)*(n^2 + 7*n + 16)/4!.
%F A063258 G.f.: (2-x)*(2-2*x+x^2)/(1-x)^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.
%F A063258 E.g.f.: (1/24)*(96 + 240*x + 120*x^2 + 20*x^3 + x^4)*exp(x). - _G. C. Greubel_, Apr 22 2024
%F A063258 a(n) = A000332(n+5)-1. - _R. J. Mathar_, Nov 22 2024
%p A063258 [seq(binomial(n+5,4)-1,n=0..37)]; # _Zerinvary Lajos_, Nov 25 2006
%t A063258 Binomial[5+Range[0,50],4] -1 (* _G. C. Greubel_, Apr 22 2024 *)
%o A063258 (PARI) { for (n=0, 1000, write("b063258.txt", n, " ", binomial(n + 5, 4) - 1) ) } \\ _Harry J. Smith_, Aug 19 2009
%o A063258 (Magma) [Binomial(n+5,4) -1 : n in [0..50]]; // _G. C. Greubel_, Apr 22 2024
%o A063258 (SageMath) [binomial(n+5,4) -1 for n in range(51)] # _G. C. Greubel_, Apr 22 2024
%Y A063258 Fifth column (r=4) of FS(4) staircase array A062750.
%Y A063258 A column of triangle A014473.
%Y A063258 Cf. A000096, A062748, A062751.
%K A063258 nonn,easy
%O A063258 0,1
%A A063258 _Wolfdieter Lang_, Jul 12 2001
%E A063258 Simpler definition from _Vladeta Jovovic_, Jul 21 2003