%I M0596 #70 Mar 31 2023 10:04:08

%S 1,1,2,3,4,10,18,28,80,162,280,880,1944,3640,12320,29160,58240,209440,

%T 524880,1106560,4188800,11022480,24344320,96342400,264539520,

%U 608608000,2504902400,7142567040,17041024000,72642169600,214277011200,528271744000,2324549427200

%N 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.

%C 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

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

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

%H T. D. Noe, <a href="/A007661/b007661.txt">Table of n, a(n) for n = 0..200</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Multifactorial.html">Multifactorial</a>.

%F a(n) = Product_{i=0..floor((n-1)/3)} (n-3*i). - _M. F. Hasler_, Feb 16 2008

%F 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

%F a(3*n) = A032031(n); a(3*n+1) = A007559(n+1); a(3*n+2) = A008544(n+1). - _Reinhard Zumkeller_, Sep 20 2013

%F 0 = a(n)*(a(n+1) -a(n+4)) +a(n+1)*a(n+3) for all n>=0. - _Michael Somos_, Feb 24 2019

%F Sum_{n>=0} 1/a(n) = A288055. - _Amiram Eldar_, Nov 10 2020

%p 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

%t 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 *)

%t RecurrenceTable[{a[0]==a[1]==1,a[2]==2,a[n]==n*a[n-3]},a,{n,30}] (* _Harvey P. Dale_, May 17 2012 *)

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

%t a[ n_] := With[{m = Mod[n, 3, 1], q = 1 + Quotient[n, 3, 1]}, If[n < 0, 0, 3^q Pochhammer[m/3, q]]]; (* _Michael Somos_, Feb 24 2019 *)

%t Table[Times@@Range[n,1,-3],{n,0,30}] (* _Harvey P. Dale_, Sep 12 2020 *)

%o (PARI) A007661(n,d=3)=prod(i=0,(n-1)\d,n-d*i) \\ _M. F. Hasler_, Feb 16 2008

%o (Haskell)

%o a007661 n k = a007661_list !! n

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

%o -- _Reinhard Zumkeller_, Sep 20 2013

%o (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

%o (Sage)

%o def a(n):

%o if (n<3): return fibonacci(n+1)

%o else: return n*a(n-3)

%o [a(n) for n in (0..30)] # _G. C. Greubel_, Aug 21 2019

%o (GAP)

%o a:= function(n)

%o if n<3 then return Fibonacci(n+1);

%o else return n*a(n-3);

%o fi;

%o end;

%o List([0..30], n-> a(n) ); # _G. C. Greubel_, Aug 21 2019

%Y Union of A007559, A008544 and A032031.

%Y Cf. A000142, A006882 (= A001147 union A000165), A007662 (= union of A007696, A001813, A008545 and A047053), A085157, A085158.

%Y Cf. A008585, A016777, A016789, A161474, A288055.

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_