A293290 -id:A293290 - cp's OEIS frontend

A324403 a(n) = Product_{i=1..n, j=1..n} (i^2 + j^2).

1, 2, 400, 121680000, 281324160000000000, 15539794609114833408000000000000, 49933566483104048708063697937367040000000000000000, 19323883089768863178599626514889213871887405416448000000000000000000000000

Offset: 0

Vaclav Kotesovec, Feb 26 2019

nonn

Next term is too long to be included.

Cf. A079478, A324426, A324437, A324438, A324439, A324440, A367834.

Cf. A272244, A293290, A324402, A324443, A324425.

a:= n-> mul(mul(i^2+j^2, i=1..n), j=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023

Table[Product[i^2+j^2, {i, 1, n}, {j, 1, n}], {n, 1, 10}]

a(n) = prod(i=1, n, prod(j=1, n, i^2+j^2)); \\ Michel Marcus, Feb 27 2019

from math import prod, factorial
def A324403(n): return (prod(i**2+j**2 for i in range(1,n) for j in range(i+1,n+1))*factorial(n))**2<Chai Wah Wu, Nov 22 2023

a(n) ~ 2^(n*(n+1) - 3/4) * exp(Pi*n*(n+1)/2 - 3*n^2 + Pi/12) * n^(2*n^2 - 1/2) / (Pi^(1/4) * Gamma(3/4)).
a(n) = 2*n^2*a(n-1)*Product_{i=1..n-1} (n^2 + i^2)^2. - Chai Wah Wu, Feb 26 2019
For n>0, a(n)/a(n-1) = A272244(n)^2 / (2*n^6). - Vaclav Kotesovec, Dec 02 2023
a(n) = exp(2*Integral_{x=0..oo} (n^2/(x*exp(x)) - (cosh(n*x) - cos(n*x))/(x*exp((n + 1)*x)*(cosh(x) - cos(x)))) dx)/2^(n^2). - Velin Yanev, Jun 30 2025

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

A203475 a(n) = Product_{1 <= i < j <= n} (i^2 + j^2).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

Each term divides its successor, as in A203476.

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

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

[(&*[(&*[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

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

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 *)

[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

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

Name edited by Alois P. Heinz, Jul 23 2017

A264596 Let S_n be the list of the first n nonnegative numbers written in binary, with least significant bits on the left, and sorted into lexicographic order; a(n) = position of n in S_n, starting indexing at 0.

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 3, 7, 1, 6, 4, 10, 3, 10, 7, 15, 1, 10, 6, 16, 4, 15, 10, 22, 3, 16, 10, 24, 7, 22, 15, 31, 1, 18, 10, 28, 6, 25, 16, 36, 4, 25, 15, 37, 10, 33, 22, 46, 3, 28, 16, 42, 10, 37, 24, 52, 7, 36, 22, 52, 15, 46, 31, 63, 1, 34, 18, 52, 10, 45, 28

Offset: 0

N. J. A. Sloane, Nov 19 2015

			S_0 = [0], a(0) = 0;
S_1 = [0, 1], a(1) = 1;
S_2 = [0, 01, 1], a(2) = 1;
S_3 = [0, 01, 1, 11], a(3) = 3;
S_4 = [0, 001, 01, 1, 11], a(4) = 1;
S_5 = [0, 001, 01, 1, 101, 11], a(5) = 4;
S_6 = [0, 001, 01, 011, 1, 101, 11], a(6) = 3;
S_7 = [0, 001, 01, 011, 1, 101, 11, 111], a(7) = 7;
S_8 = [0, 0001, 001, 01, 011, 1, 101, 11, 111], a(8) = 1;
...

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Suggested by John Bodeen's A263856.

Cf. A188215.

a:= proc(n) option remember; `if`(n=0, 0,
      `if`(irem(n, 2, 'r')=0, a(r), a(r)+r+1))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 19 2015

A264596[0]=0;A264596[n_]:=A264596[n]=A264596[Floor[n/2]]+Boole[OddQ[n]](Floor[n/2]+1);Array[A264596,100,0] (* Paolo Xausa, Nov 04 2023, after Alois P. Heinz *)

def A264596(n):
    return sorted(format(i,'b')[::-1] for i in range(n+1)).index(format(n,'b')[::-1]) # Chai Wah Wu, Nov 22 2015

a(2^n) = 1.
a(2^n-1) = 2^n-1.
a(2n) = a(n), a(2n+1) = a(n) + n+1, a(0) = 0. - Alois P. Heinz, Nov 19 2015
Conjecture: a(n) = n*(n + 3)/2 - A007814(A293290(n)) for n > 0. - Velin Yanev, Sep 12 2017

More terms from Alois P. Heinz, Nov 19 2015

A203511 a(n) = Product_{1 <= i < j <= n} (t(i) + t(j)); t = A000217 = triangular numbers.

Original entry on oeis.org

1, 1, 4, 252, 576576, 87178291200, 1386980110791475200, 3394352757964564324299571200, 1760578659300452732262852600316664217600, 255323290537547288382098619855584488593426606981120000

Offset: 0

Clark Kimberling, Jan 03 2012

nonn

Each term divides its successor, as in A203512.

See A093883 for a guide to related sequences.

Cf. A000217, A203512, A293290, A324403, A324443.

t:= n-> n*(n+1)/2:
a:= n-> mul(mul(t(i)+t(j), i=1..j-1), j=2..n):
seq(a(n), n=0..12);  # Alois P. Heinz, Jul 23 2017

f[j_] := j (j + 1)/2; z = 15;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]               (* A203511 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]  (* A203512 *)
Table[Product[k*(k+1)/2 + j*(j+1)/2, {k, 1, n}, {j, 1, k-1}], {n, 0, 10}] (* Vaclav Kotesovec, Sep 07 2023 *)

a(n) ~ c * 2^n * exp(n^2*(Pi/4 - 3/2) + n*(Pi/2 + 1)) * n^(n^2 - n - 2 - Pi/8), where c = 0.2807609661547466473998991675307759198889389396430915721129636653... - Vaclav Kotesovec, Sep 07 2023

Name edited by Alois P. Heinz, Jul 23 2017

a(0)=1 prepended by Alois P. Heinz, Jul 29 2017

A203589 Vandermonde sequence using x^2 + y^2 applied to (1,3,5,...,2n-1).

Original entry on oeis.org

1, 10, 8840, 1897064000, 192924579369600000, 15340654595434137315840000000, 1423341281300698059502838358528000000000000, 215088732628531399592688671811428988579913728000000000000000

Offset: 1

Clark Kimberling, Jan 04 2012

nonn

See A093883 for a discussion and guide to related sequences.

Cf. A293290, A324403.

f[j_] := 2 j - 1; z = 12;
v[n_] := Product[Product[f[j]^2 + f[k]^2, {j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203589 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203590 *)
Table[Product[(2*k - 1)^2 + (2*j - 1)^2, {k, 1, n}, {j, 1, k-1}], {n, 1, 10}] (* Vaclav Kotesovec, Sep 08 2023 *)

a(n) ~ 2^(3*n^2/2 - 3*n/2 - 3/8) * n^(n*(n-1)) / exp((6 - Pi)*n^2/4 - n + Pi/48). - Vaclav Kotesovec, Sep 08 2023

cp's OEIS Frontend

A324403 a(n) = Product_{i=1..n, j=1..n} (i^2 + j^2).

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A203475 a(n) = Product_{1 <= i < j <= n} (i^2 + j^2).

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A264596 Let S_n be the list of the first n nonnegative numbers written in binary, with least significant bits on the left, and sorted into lexicographic order; a(n) = position of n in S_n, starting indexing at 0.

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A203511 a(n) = Product_{1 <= i < j <= n} (t(i) + t(j)); t = A000217 = triangular numbers.

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A203589 Vandermonde sequence using x^2 + y^2 applied to (1,3,5,...,2n-1).

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