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A007531 a(n) = n*(n-1)*(n-2) (or n!/(n-3)!).
(Formerly M4159)
67
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

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

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

M. Janjic and B. Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.

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 preprint arXiv:1508.07894, 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).

FORMULA

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

a(n) = Sum(Polygorial(3, i), i = 1..n) where Polygorial(3,i) = A028896(i-1). - Daniel Dockery (peritus(AT)gmail.com), Jun 16 2003

a(n) = 3a(n-1) - 3a(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 + ... = 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 + ... + 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

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 *)

PROG

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

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

(Haskell)

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

CROSSREFS

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

Cf. A007531, A028896.

Sequence in context: A086768 A160944 A160936 * A258345 A258351 A130669

Adjacent sequences:  A007528 A007529 A007530 * A007532 A007533 A007534

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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Last modified May 23 18:42 EDT 2017. Contains 286926 sequences.