A203476 -id:A203476

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A203475

a(n) = Product_{1 <= i < j <= n} (i^2 + j^2).

+10
7

1, 5, 650, 5525000, 5807194900000, 1226800120038480000000, 77092420109247492627600000000000, 2001314057760220784660590245696000000000000000, 28468550112906756205383102673584071297339520000000000000000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Each term divides its successor, as in A203476.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..28

FORMULA

a(n) ~ c * 2^(n^2/2) * exp(Pi*n*(n+1)/4 - 3*n^2/2 + n) * n^(n*(n-1) - 3/4), where c = A323755 = sqrt(Gamma(1/4)) * exp(Pi/24) / (2*Pi)^(9/8) = 0.274528350333552903800408993482507428142383783773190451181... - Vaclav Kotesovec, Jan 26 2019

MAPLE

a:= n-> mul(mul(i^2+j^2, i=1..j-1), j=2..n):

seq(a(n), n=1..10); # Alois P. Heinz, Jul 23 2017

MATHEMATICA

f[j_]:= j^2; z = 15;

v[n_]:= Product[Product[f[k] + f[j], {j, k-1}], {k, 2, n}]

Table[v[n], {n, z}] (* A203475 *)

Table[v[n+1]/v[n], {n, z-1}] (* A203476 *)

PROG

(Magma) [(&*[(&*[j^2 + k^2: k in [1..j]])/(2*j^2): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 28 2023

(SageMath) [product(product(j^2+k^2 for k in range(1, j)) for j in range(1, n+1)) for n in range(1, 21)] # G. C. Greubel, Aug 28 2023

CROSSREFS

Cf. A000290, A093883, A202768, A203476, A272244, A293290, A323755, A324403, A367517.

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jan 02 2012

EXTENSIONS

Name edited by Alois P. Heinz, Jul 23 2017

STATUS

approved

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