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%I #56 Apr 23 2024 08:28:22
%S 3,9,19,34,55,83,119,164,219,285,363,454,559,679,815,968,1139,1329,
%T 1539,1770,2023,2299,2599,2924,3275,3653,4059,4494,4959,5455,5983,
%U 6544,7139,7769,8435,9138,9879,10659,11479,12340,13243,14189,15179,16214,17295,18423
%N Fourth column (r=3) of FS(3) staircase array A062745.
%C In the Frey-Sellers reference this sequence is called {(n+2) over 3}_{2}, n >= 0.
%C If X is an n-set and Y a fixed (n-3)-subset of X then a(n-3) is equal to the number of 3-subsets of X intersecting Y. - _Milan Janjic_, Aug 15 2007
%C Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=6, a(n-6) = coeff(charpoly(A,x), x^(n-2)). - _Milan Janjic_, Jan 26 2010
%C For n>=4, a(n-4) is the number of permutations of 1,2,...,n, such that n-3 is the only up-point, or, the same, a(n-4) is up-down coefficient {n,4} (see comment in A060351). - _Vladimir Shevelev_, Feb 14 2014
%H G. C. Greubel, <a href="/A062748/b062748.txt">Table of n, a(n) for n = 0..1000</a>
%H Guillaume Aupy and 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_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = A062745(n+2, 3) = binomial(n+4, 3) - 1 = (n+1)*(n^2 + 8*n + 18)/3!.
%F G.f.: N(3;1, x)/(1-x)^4 with N(3;1, x) = 3 - 3*x + x^2, polynomial of the second row of A062746.
%F a(n-3) = ((n^3 - n)/6) - 1, n >= 3. - _Zerinvary Lajos_, May 05 2007
%F a(n) = A000292(n+2) - 1. - _Zerinvary Lajos_, May 05 2007
%F a(n) = Sum_{i=2..n} i*(i+1)/2. - _Artur Jasinski_, Mar 14 2008
%F a(n) = -A050407(-1-n) for all n in Z. - _Michael Somos_, Jan 28 2018
%F a(n) = A000292(n+3) - A000124(n+3). - _Torlach Rush_, Aug 03 2018
%F E.g.f.: (1/6)*(18 + 36*x + 12*x^2 + x^3)*exp(x). - _G. C. Greubel_, Apr 22 2024
%e G.f. = 3 + 9*x + 19*x^2 + 34*x^3 + 55*x^4 + 83*x^5 + 119*x^6 + 164*x^7 + ...
%p seq(((n^3-n)/6)-1,n=3..40); # _Zerinvary Lajos_, May 05 2007
%t LinearRecurrence[{4,-6,4,-1},{3,9,19,34},40] (* _Harvey P. Dale_, Jan 13 2019 *)
%t Binomial[4+Range[0,50], 3] -1 (* _G. C. Greubel_, Apr 22 2024 *)
%o (PARI) {a(n) = binomial(n+4, 3) - 1}; /* _Michael Somos_, Jan 28 2018 */
%o (Magma) [Binomial(n+4,3) -1 : n in [0..50]]; // _G. C. Greubel_, Apr 22 2024
%o (SageMath) [binomial(n+4,3) - 1 for n in range(51)] # _G. C. Greubel_, Apr 22 2024
%Y A column of triangle A014473.
%Y Cf. A000124, A000292, A050407, A060351, A062745.
%K nonn,easy
%O 0,1
%A _Wolfdieter Lang_, Jul 12 2001
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