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A119688 a(n) = n!! mod (n+1).

%I A119688 #22 Feb 16 2025 08:33:01
%S A119688 1,2,3,3,3,6,1,6,5,1,3,8,7,0,1,13,9,1,15,0,11,1,9,0,13,0,7,17,15,1,1,
%T A119688 0,17,0,27,6,19,0,25,9,21,1,11,0,23,46,33,0,25,0,39,30,27,0,49,0,29,
%U A119688 58,15,50,31,0,1,0,33,1,51,0,35,1,9,27,37,0,19,0,39,78,65,0,41,82,63,0
%N A119688 a(n) = n!! mod (n+1).
%C A119688 The double factorial used here is A006882, a(n) = n*a(n-2). Bisections of A006882 are A000165 and A001147.
%H A119688 Alois P. Heinz, <a href="/A119688/b119688.txt">Table of n, a(n) for n = 1..10000</a>
%H A119688 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DoubleFactorial.html">Double Factorial</a>
%e A119688 5!! = 5*3*1 = 15, a(5) = 15 mod (5+1) = 3.
%e A119688 6!! = 6*4*2 = 48, a(6) = 48 mod (6+1) = 6.
%p A119688 P:= proc(n) local i, j, k, s; for k from 1 by 1 to n do i:=k; j:=k-2; while j >0 do i:=i*j; j:=j-2; od: s:=i mod (k+1); print(s); od: end: P(100);
%p A119688 ## another version:
%p A119688 a:= proc(n) local t, m;
%p A119688        if irem (n, 2)=1 or n<14 or isprime(n+1)
%p A119688        then t:= 1;
%p A119688             for m from n by -2 while m>1 do
%p A119688               t:= (t*m) mod (n+1)
%p A119688             od; t
%p A119688        else 0 fi
%p A119688     end:
%p A119688 seq(a(n), n=1..100);  # _Alois P. Heinz_, Feb 15 2011
%t A119688 Table[Mod[n!!,n+1],{n,100}] (* _Zak Seidov_, Feb 15 2011 *)
%o A119688 (Magma) DoubleFactorial:=func< n | &*[n..2 by -2] >; [ DoubleFactorial(n) mod (n+1): n in [1..100] ]; // _Klaus Brockhaus_, Feb 15 2011
%o A119688 (PARI) a(n) = prod(i=0, (n-1)\2, n - 2*i) % (n+1); \\ after PARI for A006882; _Michel Marcus_, Aug 22 2016
%Y A119688 Cf. A006882, A000165, A001147, A061006.
%K A119688 easy,nonn
%O A119688 1,2
%A A119688 _Paolo P. Lava_ and _Giorgio Balzarotti_, Jun 09 2006
%E A119688 a(63) corrected, a(64) inserted by _Klaus Brockhaus_, Feb 15 2011