A062748 Fourth column (r=3) of FS(3) staircase array A062745.

3, 9, 19, 34, 55, 83, 119, 164, 219, 285, 363, 454, 559, 679, 815, 968, 1139, 1329, 1539, 1770, 2023, 2299, 2599, 2924, 3275, 3653, 4059, 4494, 4959, 5455, 5983, 6544, 7139, 7769, 8435, 9138, 9879, 10659, 11479, 12340, 13243, 14189, 15179, 16214, 17295, 18423

Offset: 0

Author

Wolfdieter Lang, Jul 12 2001

Keywords

• nonn
• easy

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

Examples

			G.f. = 3 + 9*x + 19*x^2 + 34*x^3 + 55*x^4 + 83*x^5 + 119*x^6 + 164*x^7 + ...
		

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).

Crossrefs

A column of triangle A014473.

Cf. A000124, A000292, A050407, A060351, A062745.

Programs

Magma

[Binomial(n+4,3) -1 : n in [0..50]]; // G. C. Greubel, Apr 22 2024

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 *)

PARI

{a(n) = binomial(n+4, 3) - 1}; /* Michael Somos, Jan 28 2018 */

SageMath

[binomial(n+4,3) - 1 for n in range(51)] # G. C. Greubel, Apr 22 2024

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

a(n) = A000292(n+3) - A000124(n+3). - Torlach Rush, Aug 03 2018

E.g.f.: (1/6)*(18 + 36*x + 12*x^2 + x^3)*exp(x). - G. C. Greubel, Apr 22 2024