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A052762
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Products of 4 consecutive integers: a(n) = n*(n-1)*(n-2)*(n-3).
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34
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0, 0, 0, 0, 24, 120, 360, 840, 1680, 3024, 5040, 7920, 11880, 17160, 24024, 32760, 43680, 57120, 73440, 93024, 116280, 143640, 175560, 212520, 255024, 303600, 358800, 421200, 491400, 570024, 657720, 755160, 863040, 982080, 1113024
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OFFSET
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0,5
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COMMENTS
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Also, starting with n=4, the square of area of cyclic quadrilateral with sides n, n-1, n-2, n-3. - Zak Seidov, Jun 20 2003
Number of n-colorings of the complete graph on 4 vertices, which is also the tetrahedral graph. - Eric M. Schmidt, Oct 31 2012
Cf. A130534 for relations to colored forests and disposition of flags on flagpoles. - Tom Copeland, Apr 05 2014
Number of 4-permutations of the set {1, 2, ..., n}. - Joerg Arndt, Apr 05 2014
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LINKS
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FORMULA
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a(n) = n*(n-1)*(n-2)*(n-3) = n!/(n-4)! (for n >= 4).
E.g.f.: x^4*exp(x).
Recurrence: {a(1)=0, a(2)=0, a(3)=0, a(4)=24, (-1-n)*a(n) + (n-3)*a(n+1)}.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), with a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=24. - Harvey P. Dale, May 09 2012
a(n) = (n)_4 = Pochhammer(n,4), using the "falling factorial" convention; other authors write Pochhammer(x,k) for what is denoted x^(k) in the Wikipedia article, then a(n) = (n-3)^(4). - M. F. Hasler, Oct 20 2013
Sum_{n>=4} 24*(-1)^n/a(n) = A242023.
Sum_{n>=4} 1/a(n) = 1/18. (End)
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MAPLE
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spec := [S, {B=Set(Z), S=Prod(Z, Z, Z, Z, B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
G(x):=x^4*exp(x): f[0]:=G(x): for n from 1 to 34 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=0..34); # Zerinvary Lajos, Apr 05 2009
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MATHEMATICA
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Times@@@Partition[Range[-3, 60], 4, 1] (* Harvey P. Dale, May 09 2012 *)
LinearRecurrence[ {5, -10, 10, -5, 1}, {0, 0, 0, 0, 24}, 60] (* Harvey P. Dale, May 09 2012 *)
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PROG
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(Maxima) A052762(n):=n*(n-1)*(n-2)*(n-3)$
(Magma) [n*(n-1)*(n-2)*(n-3): n in [0..30]]; \\ G. C. Greubel, Nov 19 2017
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CROSSREFS
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KEYWORD
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easy,nonn,changed
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
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STATUS
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approved
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