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%I #12 Sep 08 2022 08:45:23
%S 1,1,2,6,24,120,720,10080,241920,8709120,435456000,28740096000,
%T 4828336128000,1506440871936000,759246199455744000,
%U 569434649591808000000,601322989968949248000000
%N Cumulative product of quintuple factorial A085157.
%H G. C. Greubel, <a href="/A114790/b114790.txt">Table of n, a(n) for n = 0..84</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Multifactorial.html">Multifactorial</a>.
%F a(n) = Product_{j=0..n} j!!!!!.
%F a(n) = Product_{[j=0..n} j!5.
%F a(n) = Product_{j=0..n} A085157(j).
%F a(n) = n!!!!! * a(n-1) where a(0) = 1, a(1) = 1 and n >= 2.
%F a(n) = n*(n-5)!!!!! * a(n-1) where a(0) = 1, a(1) = 1, a(2) = 2.
%e a(10) = 1!!!!! * 2!!!!! * 3!!!!! * 4!!!!! * 5!!!!! * 6!!!!! * 7!!!!! * 8!!!!! * 9!!!!! * 10!!!!! = 1 * 2 * 3 * 4 * 5 * 6 * 14 * 24 * 36 * 50 = 435456000 = 2^11 * 3^5 * 5^3 * 7.
%p b:= n-> `if`(n < 1, 1, n*b(n-5)); a:= n-> product(b(j), j = 0..n); seq(a(n), n = 0..20); # _G. C. Greubel_, Aug 21 2019
%t b[n_]:= If[n<1, 1, n*b[n-5]]; a[n_]:= Product[b[j], {j,0,n}]; Table[a[n], {n,0,20}] (* _G. C. Greubel_, Aug 21 2019 *)
%o (PARI) b(n)=if(n<1, 1, n*b(n-5));
%o vector(20, n, n--; prod(j=0,n, b(j)) ) \\ _G. C. Greubel_, Aug 21 2019
%o (Magma) b:= func< n | n eq 0 select 1 else (n lt 6) select n else n*Self(n-5) >;
%o [(&*[b(j): j in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Aug 21 2019
%o (Sage)
%o @CachedFunction
%o def b(n):
%o if (n<1): return 1
%o else: return n*b(n-5)
%o [product(b(j) for j in (0..n)) for n in (0..20)] # _G. C. Greubel_, Aug 21 2019
%o (GAP)
%o b:= function(n)
%o if n<1 then return 1;
%o else return n*b(n-5);
%o fi;
%o end;
%o List([0..20], n-> Product([0..n], j-> b(j)) ); # _G. C. Greubel_, Aug 21 2019
%Y Cf. A000178, A006882, A007662, A085157, A114347.
%K easy,nonn
%O 0,3
%A _Jonathan Vos Post_, Feb 18 2006
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