OFFSET
0,1
COMMENTS
In the Frey-Sellers reference this sequence is called {(n+2) over 3}_{2}, n >= 0.
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
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
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
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Guillaume Aupy and Julien Herrmann. Periodicity in optimal hierarchical checkpointing schemes for adjoint computations. Optimization Methods and Software, Volume 32, 2017 - Issue 3. Preprint
D. D. Frey and J. A. Sellers, Generalizing Bailey's generalization of the Catalan numbers, The Fibonacci Quarterly, 39 (2001) 142-148.
Milan Janjic, Two Enumerative Functions
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = A062745(n+2, 3) = binomial(n+4, 3) - 1 = (n+1)*(n^2 + 8*n + 18)/3!.
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.
a(n-3) = ((n^3 - n)/6) - 1, n >= 3. - Zerinvary Lajos, May 05 2007
a(n) = A000292(n+2) - 1. - Zerinvary Lajos, May 05 2007
a(n) = Sum_{i=2..n} i*(i+1)/2. - Artur Jasinski, Mar 14 2008
a(n) = -A050407(-1-n) for all n in Z. - Michael Somos, Jan 28 2018
E.g.f.: (1/6)*(18 + 36*x + 12*x^2 + x^3)*exp(x). - G. C. Greubel, Apr 22 2024
EXAMPLE
G.f. = 3 + 9*x + 19*x^2 + 34*x^3 + 55*x^4 + 83*x^5 + 119*x^6 + 164*x^7 + ...
MAPLE
seq(((n^3-n)/6)-1, n=3..40); # Zerinvary Lajos, May 05 2007
MATHEMATICA
LinearRecurrence[{4, -6, 4, -1}, {3, 9, 19, 34}, 40] (* Harvey P. Dale, Jan 13 2019 *)
Binomial[4+Range[0, 50], 3] -1 (* G. C. Greubel, Apr 22 2024 *)
PROG
(PARI) {a(n) = binomial(n+4, 3) - 1}; /* Michael Somos, Jan 28 2018 */
(Magma) [Binomial(n+4, 3) -1 : n in [0..50]]; // G. C. Greubel, Apr 22 2024
(SageMath) [binomial(n+4, 3) - 1 for n in range(51)] # G. C. Greubel, Apr 22 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jul 12 2001
STATUS
approved