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A085158 Sextuple factorials, 6-factorials, n!!!!!!, n!6.

%I A085158 #21 Feb 16 2025 08:32:50
%S A085158 1,1,2,3,4,5,6,7,16,27,40,55,72,91,224,405,640,935,1296,1729,4480,
%T A085158 8505,14080,21505,31104,43225,116480,229635,394240,623645,933120,
%U A085158 1339975,3727360,7577955,13404160,21827575,33592320,49579075,141639680
%N A085158 Sextuple factorials, 6-factorials, n!!!!!!, n!6.
%C A085158 The term "Sextuple factorial numbers" is also used for the sequences A008542, A008543, A011781, A047058, A047657, A049308, which have a different definition. The definition given here is the one commonly used.
%H A085158 G. C. Greubel, <a href="/A085158/b085158.txt">Table of n, a(n) for n = 0..1000</a>
%H A085158 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Multifactorial.html">Multifactorial</a>.
%F A085158 a(n)=1 for n < 1, otherwise a(n) = n*a(n-6).
%F A085158 Sum_{n>=0} 1/a(n) = A288093. - _Amiram Eldar_, Nov 10 2020
%e A085158 a(14) = 224 because 14*a(14-6) = 14*a(8) = 14*16 = 224.
%p A085158 a:= n-> `if`(n<1, 1, n*a(n-6)); seq(a(n), n=0..40); # _G. C. Greubel_, Aug 21 2019
%t A085158 Table[Times@@Range[n,1,-6],{n,0,40}] (* _Harvey P. Dale_, Aug 10 2019 *)
%o A085158 (PARI) a(n)=if(n<1, 1, n*a(n-6));
%o A085158 vector(40, n, n--; a(n) ) \\ _G. C. Greubel_, Aug 21 2019
%o A085158 (Magma) b:=func< n | n le 6 select n else n*Self(n-6) >;
%o A085158 [1] cat [b(n): n in [1..40]]; // _G. C. Greubel_, Aug 21 2019
%o A085158 (Sage)
%o A085158 def a(n):
%o A085158     if (n<1): return 1
%o A085158     else: return n*a(n-6)
%o A085158 [a(n) for n in (0..40)] # _G. C. Greubel_, Aug 21 2019
%o A085158 (GAP)
%o A085158 a:= function(n)
%o A085158     if n<1 then return 1;
%o A085158     else return n*a(n-6);
%o A085158     fi;
%o A085158   end;
%o A085158 List([0..40], n-> a(n) ); # _G. C. Greubel_, Aug 21 2019
%Y A085158 Cf. n!:A000142, n!!:A006882, n!!!:A007661, n!!!!:A007662, n!!!!!:A085157, 6-factorial primes: n!!!!!!+1:A085150, n!!!!!!-1:A051592.
%Y A085158 Cf. A288093.
%K A085158 nonn
%O A085158 0,3
%A A085158 _Hugo Pfoertner_, Jun 21 2003