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A007661 Triple factorial numbers a(n) = n!!!, defined by a(n) = n*a(n-3), a(0) = a(1) = 1, a(2) = 2. Sometimes written n!3.
(Formerly M0596)
65
1, 1, 2, 3, 4, 10, 18, 28, 80, 162, 280, 880, 1944, 3640, 12320, 29160, 58240, 209440, 524880, 1106560, 4188800, 11022480, 24344320, 96342400, 264539520, 608608000, 2504902400, 7142567040, 17041024000, 72642169600 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The triple factorial of a positive integer n is the product of the positive integers <= n that have the same residue modulo 3 as n. - Peter Luschny, Jun 23 2011

a(3*n) = A032031(n); a(3*n+1) = A007559(n+1); a(3*n+2) = A008544(n+1). - Reinhard Zumkeller, Sep 20 2013

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. Spanier and K. B. Oldham, An Atlas of Functions, Hemisphere, NY, 1987, p. 23.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..200

Eric Weisstein's World of Mathematics, Multifactorial.

FORMULA

a(n) = prod(i=0..floor((n-1)/3), n-3*i ). - M. F. Hasler, Feb 16 2008

a(n) ~ c * n^(n/3+1/2)/exp(n/3), where c = sqrt(2*Pi/3) if n=3*k, c = sqrt(2*Pi)*3^(1/6) / Gamma(1/3) if n=3*k+1, c = sqrt(2*Pi)*3^(-1/6) / Gamma(2/3) if n=3*k+2. - Vaclav Kotesovec, Jul 29 2013

MAPLE

A007661 := n -> mul(k, k = select(k -> k mod 3 = n mod 3, [$1 .. n])): seq(A007661(n), n = 0 .. 29);  # Peter Luschny, Jun 23 2011

MATHEMATICA

multiFactorial[n_, k_] := If[n < 1, 1, If[n < k + 1, n, n*multiFactorial[n - k, k]]]; Array[ multiFactorial[#, 3] &, 30, 0] (* Robert G. Wilson v, Apr 23 2011 *)

RecurrenceTable[{a[0]==a[1]==1, a[2]==2, a[n]==n*a[n-3]}, a, {n, 30}] (* Harvey P. Dale, May 17 2012 *)

Table[With[{q = Quotient[n + 2, 3]}, 3^q q! Binomial[n/3, q]], {n, 0, 30}] (* Jan Mangaldan, Mar 21 2013 *)

PROG

(PARI) a(n, d=3)=prod(i=0, (n-1)\d, n-d*i) \\ M. F. Hasler, Feb 16 2008

(PARI) a(n) = prod(i=0, floor((n-1)/3), n-3*i );

(Haskell)

a007661 n k = a007661_list !! n

a007661_list = 1 : 1 : 2 : zipWith (*) a007661_list [3..]

-- Reinhard Zumkeller, Sep 20 2013

(MAGMA) I:=[1, 1, 2]; [n le 3 select I[n] else (n-1)*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Nov 27 2015

CROSSREFS

Cf. A000142, A006882 (= A001147 union A000165), A007662.

Cf. A008585, A016777, A016789.

Sequence in context: A055506 A098088 A080500 * A049891 A135432 A108364

Adjacent sequences:  A007658 A007659 A007660 * A007662 A007663 A007664

KEYWORD

nonn,easy,nice

AUTHOR

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

STATUS

approved

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Last modified April 27 10:54 EDT 2017. Contains 285512 sequences.