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

1, 1, 2, 3, 4, 5, 6, 14, 24, 36, 50, 66, 168, 312, 504, 750, 1056, 2856, 5616, 9576, 15000, 22176, 62832, 129168, 229824, 375000, 576576, 1696464, 3616704, 6664896, 11250000, 17873856, 54286848, 119351232, 226606464, 393750000, 643458816

Offset: 0

Author

Hugo Pfoertner, Jun 21 2003

Keywords

• nonn

Comments

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.

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Peter Luschny, Multifactorials
• Eric Weisstein's World of Mathematics, Multifactorial.

Crossrefs

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

Cf. A288092.

Programs

GAP

a:= function(n)
    if n<1 then return 1;
    else return n*a(n-5);
    fi;
  end;
List([0..40], n-> a(n) ); # G. C. Greubel, Aug 18 2019

Magma

b:= func< n | (n lt 6) select n else n*Self(n-5) >;
[1] cat [b(n): n in [1..40]]; // G. C. Greubel, Aug 18 2019

Maple

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

Mathematica

a[n_]:= If[n<1, 1, n*a[n-5]]; Table[a[n], {n,0,40}] (* G. C. Greubel, Aug 18 2019 *)
Table[Times@@Range[n,1,-5],{n,0,40}] (* Harvey P. Dale, May 12 2020 *)

PARI

a(n)=if(n<1, 1, n*a(n-5))
for(n=0,50,print1(a(n),",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 19 2006

Python

def A085157(n):
    if n <= 0:
        return 1
    else:
        return n*A085157(n-5)
n = 0
while n <= 40:
    print(n,A085157(n))
    n = n+1 # A.H.M. Smeets, Aug 18 2019

Sage

def a(n):
    if (n<1): return 1
    else: return n*a(n-5)
[a(n) for n in (0..40)] # G. C. Greubel, Aug 18 2019

Formula

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

Sum_{n>=0} 1/a(n) = A288092. - Amiram Eldar, Nov 10 2020