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Quintuple factorials, 5-factorials, n!!!!!, n!5.
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%I #31 Sep 08 2022 08:45:11

%S 1,1,2,3,4,5,6,14,24,36,50,66,168,312,504,750,1056,2856,5616,9576,

%T 15000,22176,62832,129168,229824,375000,576576,1696464,3616704,

%U 6664896,11250000,17873856,54286848,119351232,226606464,393750000,643458816

%N Quintuple factorials, 5-factorials, n!!!!!, n!5.

%C The term "Quintuple factorial numbers" is also used for the sequences A008546, A008548, A052562, A047055, A047056 which have a different definition. The definition given here is the one commonly used.

%H G. C. Greubel, <a href="/A085157/b085157.txt">Table of n, a(n) for n = 0..1000</a>

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/Multifactorials">Multifactorials</a>

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

%F a(n) = 1 for n < 1, otherwise a(n) = n*a(n-5).

%F Sum_{n>=0} 1/a(n) = A288092. - _Amiram Eldar_, Nov 10 2020

%e a(12) = 168 because 12*a(12-5) = 12*a(7) = 12*14 = 168.

%p a:= n-> `if`(n < 1, 1, n*a(n-5)) end proc; seq(a(n), n = 0..40); # _G. C. Greubel_, Aug 18 2019

%t a[n_]:= If[n<1, 1, n*a[n-5]]; Table[a[n], {n,0,40}] (* _G. C. Greubel_, Aug 18 2019 *)

%t Table[Times@@Range[n,1,-5],{n,0,40}] (* _Harvey P. Dale_, May 12 2020 *)

%o (PARI) a(n)=if(n<1, 1, n*a(n-5))

%o for(n=0,50,print1(a(n),",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 19 2006

%o (Magma)

%o b:= func< n | (n lt 6) select n else n*Self(n-5) >;

%o [1] cat [b(n): n in [1..40]]; // _G. C. Greubel_, Aug 18 2019

%o (Sage)

%o def a(n):

%o if (n<1): return 1

%o else: return n*a(n-5)

%o [a(n) for n in (0..40)] # _G. C. Greubel_, Aug 18 2019

%o (GAP)

%o a:= function(n)

%o if n<1 then return 1;

%o else return n*a(n-5);

%o fi;

%o end;

%o List([0..40], n-> a(n) ); # _G. C. Greubel_, Aug 18 2019

%o (Python)

%o def A085157(n):

%o if n <= 0:

%o return 1

%o else:

%o return n*A085157(n-5)

%o n = 0

%o while n <= 40:

%o print(n,A085157(n))

%o n = n+1 # _A.H.M. Smeets_, Aug 18 2019

%Y Cf. n!:A000142, n!!:A006882, n!!!:A007661, n!!!!:A007662, n!!!!!!:A085158, 5-factorial primes: n!!!!!+1:A085148, n!!!!!-1:A085149.

%Y Cf. A288092.

%K nonn

%O 0,3

%A _Hugo Pfoertner_, Jun 21 2003