login
a(n) = n^2*(n-2)^2*(n-4)^2*...*(1 or 2)^2.
7

%I #48 May 14 2023 15:42:09

%S 1,1,4,9,64,225,2304,11025,147456,893025,14745600,108056025,

%T 2123366400,18261468225,416179814400,4108830350625,106542032486400,

%U 1187451971330625,34519618525593600,428670161650355625,13807847410237440000,189043541287806830625,6682998146554920960000,100004033341249813400625

%N a(n) = n^2*(n-2)^2*(n-4)^2*...*(1 or 2)^2.

%H David A. Corneth, <a href="/A184877/b184877.txt">Table of n, a(n) for n = 0..449</a>

%F a(n) = (n!!)^2 = A006882(n)^2. - _Gionata Neri_, Oct 29 2015

%F For n > 1, a(n) = n^2 * a(n-2). - _David A. Corneth_, Aug 03 2017

%F From _Amiram Eldar_, Apr 09 2022: (Start)

%F Sum_{n>=0} 1/a(n) = BesselI(0, 1) + StruveL(0, 1)*Pi/2 = A197036 + A197037 * Pi/2.

%F Sum_{n>=0} (-1)^n/a(n) = BesselI(0, 1) - StruveL(0, 1)*Pi/2. (End)

%F E.g.f.: 1/(1-x^2) + x*(1+arcsin(x))/(1-x^2)^(3/2). - _Fabián Pereyra_, May 14 2023

%e a(0) = Empty product = 1;

%e a(1) = 1^2 = 1;

%e a(2) = 2^2 = 4;

%e a(3) = 3^2*1^2 = 9;

%e a(4) = 4^2*2^2 = 64;

%e a(5) = 5^2*3^2*1^2 = 225;

%e ...

%t Table[Product[(n-2*k)^2, {k,0,Floor[(n-1)/2]}], {n,0,50}] (* _G. C. Greubel_, Oct 14 2018 *)

%o (PARI) vector(100, n, n--; prod(k=0, (n-1)\2, (n-2*k)^2)) \\ _Altug Alkan_, Oct 29 2015

%o (PARI) first(n) = {if(n<2, return(vector(n, i, 1))); my(res = vector(n), i = 3); res[1] = res[2] = 1; while(i<=n, res[i] = res[i-2]*(i-1)^2; i++) ;res} \\ _David A. Corneth_, Aug 03 2017

%o (Magma) [1] cat [(&*[(n-2*k)^2: k in [0..Floor((n-1)/2)]]): n in [1..50]]; // _G. C. Greubel_, Oct 14 2018

%Y Rightmost diagonal of A182971.

%Y With signs, a row of A288580.

%Y Cf. A006882, A197036, A197037.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Feb 01 2011