A184877 a(n) = n^2*(n-2)^2*(n-4)^2*...*(1 or 2)^2.

1, 1, 4, 9, 64, 225, 2304, 11025, 147456, 893025, 14745600, 108056025, 2123366400, 18261468225, 416179814400, 4108830350625, 106542032486400, 1187451971330625, 34519618525593600, 428670161650355625, 13807847410237440000, 189043541287806830625, 6682998146554920960000, 100004033341249813400625

Offset: 0

Author

N. J. A. Sloane, Feb 01 2011

Keywords

• nonn

Examples

			a(0) = Empty product = 1;
a(1) = 1^2 = 1;
a(2) = 2^2 = 4;
a(3) = 3^2*1^2 = 9;
a(4) = 4^2*2^2 = 64;
a(5) = 5^2*3^2*1^2 = 225;
...
		

Links

• David A. Corneth, Table of n, a(n) for n = 0..449

Crossrefs

Rightmost diagonal of A182971.

With signs, a row of A288580.

Cf. A006882, A197036, A197037.

Programs

Magma

[1] cat [(&*[(n-2*k)^2: k in [0..Floor((n-1)/2)]]): n in [1..50]]; // G. C. Greubel, Oct 14 2018

Mathematica

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

PARI

vector(100, n, n--; prod(k=0, (n-1)\2, (n-2*k)^2)) \\ Altug Alkan, Oct 29 2015

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

Formula

a(n) = (n!!)^2 = A006882(n)^2. - Gionata Neri, Oct 29 2015

For n > 1, a(n) = n^2 * a(n-2). - David A. Corneth, Aug 03 2017

From Amiram Eldar, Apr 09 2022: (Start)

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

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

E.g.f.: 1/(1-x^2) + x*(1+arcsin(x))/(1-x^2)^(3/2). - Fabián Pereyra, May 14 2023