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A265694 a(n) = n!! mod n^2 where n!! is a double factorial number (A006882).

%I A265694 #19 Sep 08 2022 08:46:15
%S A265694 0,2,3,8,15,12,7,0,54,40,110,0,104,84,0,0,221,0,342,0,0,220,506,0,0,
%T A265694 312,0,0,493,0,930,0,0,544,0,0,222,684,0,0,369,0,1806,0,0,1012,47,0,0,
%U A265694 0,0,0,1590,0,0,0,0,1624,59,0,3050,1860,0,0,0,0,4422,0,0,0
%N A265694 a(n) = n!! mod n^2 where n!! is a double factorial number (A006882).
%C A265694 Inspired by geometric meaning of distribution of 0's in this sequence.
%C A265694 Position of 0's in this sequence is directly related with sequence which gives the short leg of more than one Pythagorean triangle (A009188). See comment sections in A009188 and A264828 which are the related sequences for further information.
%C A265694 More precisely, a(A009188(n+1)) = 0 for n > 0.
%F A265694 a(A009188(n+1)) = 0 for n > 0.
%e A265694 For n = 1, a(1) = 1!! mod 1^2 = 1 mod 1 = 0.
%e A265694 For n = 2, a(2) = 2!! mod 2^2 = 2 mod 4 = 2.
%e A265694 For n = 8, a(8) = 8!! mod 8^2 = 384 mod 64 = 0.
%t A265694 Table[Mod[n!!, n^2], {n, 70}] (* _Michael De Vlieger_, Dec 14 2015 *)
%o A265694 (PARI) df(n) = if( n<0, 0, my(E); E = exp(x^2 / 2 + x * O(x^n)); n! * polcoeff( 1 + E * x * (1 + intformal(1 / E)), n));
%o A265694 vector(70, n, df(n) % n^2)
%o A265694 (Magma) DoubleFactorial:=func< n | &*[n..2 by -2] >; [ DoubleFactorial(n) mod n^2: n in [1..70] ]; // _Vincenzo Librandi_, Dec 14 2015
%Y A265694 Cf. A006882.
%K A265694 nonn,easy
%O A265694 1,2
%A A265694 _Altug Alkan_, Dec 13 2015