A007531 a(n) = n*(n-1)*(n-2) (or n!/(n-3)!).

0, 0, 0, 6, 24, 60, 120, 210, 336, 504, 720, 990, 1320, 1716, 2184, 2730, 3360, 4080, 4896, 5814, 6840, 7980, 9240, 10626, 12144, 13800, 15600, 17550, 19656, 21924, 24360, 26970, 29760, 32736, 35904, 39270, 42840, 46620, 50616, 54834, 59280, 63960, 68880

Offset: 0

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

• nonn
• easy

Comments

Ed Pegg Jr conjectures that n^3 - n = k! has a solution if and only if n is 2, 3, 5 or 9 (when k is 3, 4, 5 and 6).

Three-dimensional promic (or oblong) numbers, cf. A002378. - Alexandre Wajnberg, Dec 29 2005

Doubled first differences of tritriangular numbers A050534(n) = (1/8)n(n + 1)(n - 1)(n - 2). a(n) = 2*(A050534(n+1) - A050534(n)). - Alexander Adamchuk, Apr 11 2006

If Y is a 4-subset of an n-set X then, for n >= 6, a(n-4) is the number of (n-5)-subsets of X having exactly two elements in common with Y. - Milan Janjic, Dec 28 2007

Convolution of A005843 with A008585. - Reinhard Zumkeller, Mar 07 2009

a(n) = A000578(n) - A000567(n). - Reinhard Zumkeller, Sep 18 2009

For n > 3: a(n) = A173333(n, n-3). - Reinhard Zumkeller, Feb 19 2010

Let H be the n X n Hilbert matrix H(i, j) = 1/(i+j-1) for 1 <= i, j <= n. Let B be the inverse matrix of H. The sum of the elements in row 2 of B equals (-1)^n a(n+1). - T. D. Noe, May 01 2011

a(n) equals 2^(n-1) times the coefficient of log(3) in 2F1(n-2, n-2, n, -2). - John M. Campbell, Jul 16 2011

For n > 2 a(n) = 1/(Integral_{x = 0..Pi/2} (sin(x))^5*(cos(x))^(2*n-5)). - Francesco Daddi, Aug 02 2011

a(n) is the number of functions f:[3] -> [n] that are injective since there are n choices for f(1), (n-1) choices for f(2), and (n-2) choices for f(3). Also, a(n+1) is the number of functions f:[3] -> [n] that are width-2 restricted (that is, the pre-image under f of any element in [n] is of size 2 or less). See "Width-restricted finite functions" link below. - Dennis P. Walsh, Mar 01 2012

This sequence is produced by three consecutive triangular numbers t(n-1), t(n-2) and t(n-3) in the expression 2*t(n-1)*(t(n-2)-t(n-3)) for n = 0, 1, 2, ... - J. M. Bergot, May 14 2012

For n > 2: A020639(a(n)) = 2; A006530(a(n)) = A093074(n-1). - Reinhard Zumkeller, Jul 04 2012

Number of contact points between equal spheres arranged in a tetrahedron with n - 1 spheres in each edge. - Ignacio Larrosa Cañestro, Jan 07 2013

Also for n >= 3, area of Pythagorean triangle in which one side differs from hypotenuse by two units. Consider any Pythagorean triple (2n, n^2-1, n^2+1) where n > 1. The area of such a Pythagorean triangle is n(n^2-1). For n = 2, 3, 4,.. the areas are 6, 24, 60, .... which are the given terms of the series. - Jayanta Basu, Apr 11 2013

Cf. A130534 for relations to colored forests, disposition of flags on flagpoles, and colorings of the vertices (chromatic polynomial) of the complete graph K_3. - Tom Copeland, Apr 05 2014

Starting with 6, 24, 60, 120, ..., a(n) is the number of permutations of length n>=3 avoiding the partially ordered pattern (POP) {1>2} of length 5. That is, the number of length n permutations having no subsequences of length 5 in which the first element is larger than the second element. - Sergey Kitaev, Dec 11 2020

For integer m and positive integer r >= 2, the polynomial a(n) + a(n + m) + a(n + 2*m) + ... + a(n + r*m) in n has its zeros on the vertical line Re(n) = (2 - r*m)/2 in the complex plane. - Peter Bala, Jun 02 2024

References

• R. K. Guy, Unsolved Problems in Theory of Numbers, Section D25.
• L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 40.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• G. D. Birkhoff, A determinant formula for the number of ways of coloring a map, Ann. Math., 14:42-4. See 1st polynomial p. 5.
• Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
• Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
• Milan Janjic and B. Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
• Michael Penn, A natural nested root, YouTube video, 2022.
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv:1406.3081 [math.CO], 2014-2015.
• Michelle Rudolph-Lilith, On the Product Representation of Number Sequences, with Application to the Fibonacci Family, arXiv:1508.07894 [math.NT], 2015.
• Dennis Walsh, Width-restricted finite functions
• Index entries for two-way infinite sequences
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

binomial(n, k): A161680 (k = 2), A000332 (k = 4), A000389 (k = 5), A000579 (k = 6), A000580 (k = 7), A000581 (k = 8), A000582 (k = 9).

Cf. A002378, A005563, A084939, A084940, A084941, A084942, A084943, A084944.

Cf. A028896.

Programs

Haskell

a007531 n = product [n-2..n]  -- Reinhard Zumkeller, Jul 04 2012

Magma

[n*(n-1)*(n-2): n in [0..40]]; // Vincenzo Librandi, May 02 2011

Maple

[seq(6*binomial(n,3),n=0..41)]; # Zerinvary Lajos, Nov 24 2006

Mathematica

Table[n^3 - 3n^2 + 2n, {n, 0, 42}]
Table[FactorialPower[n, 3], {n, 0, 42}] (* Arkadiusz Wesolowski, Oct 29 2012 *)

PARI

a(n)=n*(n-1)*(n-2)

Sage

[n*(n-1)*(n-2) for n in range(40)] # G. C. Greubel, Feb 11 2019

Formula

a(n) = 6*A000292(n-2).

a(n) = Sum_{i=1..n} polygorial(3,i) where polygorial(3,i) = A028896(i-1). - Daniel Dockery (peritus(AT)gmail.com), Jun 16 2003

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 6, n > 2. - Zak Seidov, Feb 09 2006

G.f.: 6*x^2/(1-x)^4.

a(-n) = -a(n+2).

1/6 + 3/24 + 5/60 + ... = Sum_{k>=1} (2*k-1)/(k*(k+1)*(k+2)) = 3/4. [Jolley Eq. 213]

a(n+1) = n^3 - n. - Mohammad K. Azarian, Jul 26 2007

E.g.f.: x^3*exp(x). - Geoffrey Critzer, Feb 08 2009

If the first 0 is eliminated, a(n) = floor(n^5/(n^2+1)). - Gary Detlefs, Feb 11 2010

1/6 + 1/24 + 1/60 + ... = Sum_{n>=1} 1/(n*(n+1)*(n+2)) = 1/4. - Mohammad K. Azarian, Dec 29 2010

a(0) = 0, a(n) = a(n-1) + 3*(n-1)*(n-2). - Jean-François Alcover, Jan 08 2013

(a(n+1) - a(n))/6 = A000217(n-2) for n > 0. - J. M. Bergot, Jul 30 2013

Partial sums of A028896. - R. J. Mathar, Aug 28 2014

1/6 + 1/24 + 1/60 + ... + 1/(n*(n+1)*(n+2)) = n*(n+3)/(4*(n+1)*(n+2)). - Christina Steffan, Jul 20 2015

a(n+2)^2 = A005563(n)^3 + A005563(n)^2. - Bruno Berselli, May 03 2018

a(n)*a(n+1) + A000096(n-3)^2 = m^2 (a perfect square), m = ((a(n)+a(n+1))/2)-n. - Ezhilarasu Velayutham, May 21 2019

Sum_{n>=3} (-1)^(n+1)/a(n) = 2*log(2) - 5/4. - Amiram Eldar, Jul 02 2020

For n >= 3, (a(n) + (a(n) + (a(n) + ...)^(1/3))^(1/3))^(1/3) = n - 1. - Paolo Xausa, Apr 09 2022