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

4, 14, 34, 69, 125, 209, 329, 494, 714, 1000, 1364, 1819, 2379, 3059, 3875, 4844, 5984, 7314, 8854, 10625, 12649, 14949, 17549, 20474, 23750, 27404, 31464, 35959, 40919, 46375, 52359, 58904, 66044, 73814, 82250, 91389, 101269, 111929, 123409, 135750

Offset: 0

Author

Wolfdieter Lang, Jul 12 2001

Keywords

• nonn
• easy

Comments

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

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

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

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000
• Guillaume Aupy, 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 (5,-10,10,-5,1).

Crossrefs

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

A column of triangle A014473.

Cf. A000096, A062748, A062751.

Programs

Magma

[Binomial(n+5,4) -1 : n in [0..50]]; // G. C. Greubel, Apr 22 2024

Maple

[seq(binomial(n+5,4)-1,n=0..37)]; # Zerinvary Lajos, Nov 25 2006

Mathematica

Binomial[5+Range[0,50],4] -1 (* G. C. Greubel, Apr 22 2024 *)

PARI

{ for (n=0, 1000, write("b063258.txt", n, " ", binomial(n + 5, 4) - 1) ) } \\ Harry J. Smith, Aug 19 2009

SageMath

[binomial(n+5,4) -1 for n in range(51)] # G. C. Greubel, Apr 22 2024

Formula

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

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.

E.g.f.: (1/24)*(96 + 240*x + 120*x^2 + 20*x^3 + x^4)*exp(x). - G. C. Greubel, Apr 22 2024

a(n) = A000332(n+5)-1. - R. J. Mathar, Nov 22 2024

Extensions

Simpler definition from Vladeta Jovovic, Jul 21 2003