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A085157 Quintuple factorials, 5-factorials, n!!!!!, n!5.

%I A085157 #32 Feb 16 2025 08:32:50
%S A085157 1,1,2,3,4,5,6,14,24,36,50,66,168,312,504,750,1056,2856,5616,9576,
%T A085157 15000,22176,62832,129168,229824,375000,576576,1696464,3616704,
%U A085157 6664896,11250000,17873856,54286848,119351232,226606464,393750000,643458816
%N A085157 Quintuple factorials, 5-factorials, n!!!!!, n!5.
%C A085157 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 A085157 G. C. Greubel, <a href="/A085157/b085157.txt">Table of n, a(n) for n = 0..1000</a>
%H A085157 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/Multifactorials">Multifactorials</a>
%H A085157 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Multifactorial.html">Multifactorial</a>.
%F A085157 a(n) = 1 for n < 1, otherwise a(n) = n*a(n-5).
%F A085157 Sum_{n>=0} 1/a(n) = A288092. - _Amiram Eldar_, Nov 10 2020
%e A085157 a(12) = 168 because 12*a(12-5) = 12*a(7) = 12*14 = 168.
%p A085157 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 A085157 a[n_]:= If[n<1, 1, n*a[n-5]]; Table[a[n], {n,0,40}] (* _G. C. Greubel_, Aug 18 2019 *)
%t A085157 Table[Times@@Range[n,1,-5],{n,0,40}] (* _Harvey P. Dale_, May 12 2020 *)
%o A085157 (PARI) a(n)=if(n<1, 1, n*a(n-5))
%o A085157 for(n=0,50,print1(a(n),",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 19 2006
%o A085157 (Magma)
%o A085157 b:= func< n | (n lt 6) select n else n*Self(n-5) >;
%o A085157 [1] cat [b(n): n in [1..40]]; // _G. C. Greubel_, Aug 18 2019
%o A085157 (Sage)
%o A085157 def a(n):
%o A085157     if (n<1): return 1
%o A085157     else: return n*a(n-5)
%o A085157 [a(n) for n in (0..40)] # _G. C. Greubel_, Aug 18 2019
%o A085157 (GAP)
%o A085157 a:= function(n)
%o A085157     if n<1 then return 1;
%o A085157     else return n*a(n-5);
%o A085157     fi;
%o A085157   end;
%o A085157 List([0..40], n-> a(n) ); # _G. C. Greubel_, Aug 18 2019
%o A085157 (Python)
%o A085157 def A085157(n):
%o A085157     if n <= 0:
%o A085157         return 1
%o A085157     else:
%o A085157         return n*A085157(n-5)
%o A085157 n = 0
%o A085157 while n <= 40:
%o A085157     print(n,A085157(n))
%o A085157     n = n+1 # _A.H.M. Smeets_, Aug 18 2019
%Y A085157 Cf. n!:A000142, n!!:A006882, n!!!:A007661, n!!!!:A007662, n!!!!!!:A085158, 5-factorial primes: n!!!!!+1:A085148, n!!!!!-1:A085149.
%Y A085157 Cf. A288092.
%K A085157 nonn
%O A085157 0,3
%A A085157 _Hugo Pfoertner_, Jun 21 2003