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%I #12 Sep 08 2022 08:45:23
%S 1,2,4,7,11,16,22,36,60,96,146,212,380,692,1196,1946,3002,5858,11474,
%T 21050,36050,58226,121058,250226,480050,855050,1431626,3128090,
%U 6744794,13409690,24659690,42533546,96820394,216171626,442778090,836528090
%N Cumulative sum of quintuple factorial numbers n!!!!! (A085157).
%C a(1) = 2 is prime; a(3) = 7 is prime; a(4) = 11 is prime; and there are no more primes in the sequence. Semiprime values are: a(2) = 4 = 2^2, a(6) = 22, a(10) = 146 = 2 * 73, a(18) = 11474 = 2 * 5737, a(23) = 250226 = 2 * 125113.
%H G. C. Greubel, <a href="/A114805/b114805.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{j=0..n} j!5.
%F a(n) = Sum_{j=0..n} j!!!!!.
%F a(n) = Sum_{j=0..n} A085157(j).
%e a(10) = 0!5 + 1!5 + 2!5 + 3!5 + 4!5 + 5!5 + 6!5 + 7!5 + 8!5 + 9!5 + 10!5 =
%e 1 + 1 + 2 + 3 + 4 + 5 + 6 + 14 + 24 + 36 + 50 = 146 = 2 * 73.
%p b:= n-> `if`(n < 1, 1, n*b(n-5)); a:= n-> sum(b(j), j = 0..n); seq(a(n), n = 0..40); # _G. C. Greubel_, Aug 21 2019
%t f5[0]=1; f5[n_]:= f5[n]= If[n<=6, n, n f5[n-5]]; Accumulate[f5/@Range[0, 35]] (* _Giovanni Resta_, Jun 15 2016 *)
%o (PARI) b(n)=if(n<1, 1, n*b(n-5));
%o vector(40, n, n--; sum(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..40]]; // _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 [sum(b(j) for j in (0..n)) for n in (0..40)] # _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..40], n-> Sum([0..n], j-> b(j)) ); # _G. C. Greubel_, Aug 21 2019
%Y Cf. A007662, A085157, A114347.
%K easy,nonn
%O 0,2
%A _Jonathan Vos Post_, Feb 18 2006
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