A324437 -id:A324437 - cp's OEIS frontend

A306620 Decimal expansion of a constant related to the asymptotics of A324437.

2, 3, 4, 5, 1, 5, 8, 4, 4, 5, 1, 4, 0, 4, 2, 7, 9, 2, 8, 1, 8, 0, 7, 1, 4, 3, 3, 1, 7, 5, 0, 0, 5, 1, 8, 6, 6, 0, 6, 9, 6, 2, 9, 3, 9, 4, 4, 9, 6, 1, 0, 3, 9, 5, 5, 3, 2, 4, 5, 8, 2, 3, 6, 8, 3, 6, 6, 1, 0, 9, 9, 4, 1, 7, 0, 2, 5, 3, 0, 3, 2, 4, 1, 6, 1, 4, 5, 1, 7, 7, 7, 4, 7, 0, 5, 4, 3, 0, 2, 6, 0, 4, 9, 6, 6, 0

Offset: 0

Vaclav Kotesovec, Mar 01 2019

			0.234515844514042792818071433175005186606962939449610395532458236836610994170253...

Vaclav Kotesovec, Table of n, a(n) for n = 0..158

Cf. A324437, A367834.

Equals limit_{n->oo} A324437(n) / (2^(n*(n+1)) * exp(Pi*n*(n+1)/sqrt(2) - 6*n^2) * (1 + sqrt(2))^(sqrt(2)*n*(n+1)) * n^(4*n^2 - 1)).

Equals limit_{n->oo} n*(Product_{i=1..n, j=1..n} ((i/n)^4 + (j/n)^4)) / exp(6*n + n*(n+1)*Integral_{x=0..1, y=0..1} log(x^4 + y^4) dy dx). - Vaclav Kotesovec, Dec 04 2023

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

Original entry on oeis.org

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

A324426 a(n) = Product_{i=1..n, j=1..n} (i^3 + j^3).

Original entry on oeis.org

1, 2, 2592, 134425267200, 3120795915109442519040000, 180825857777547616919759624941086965760000000, 99356698720512072045648926659510730227553351200000695922065408000000000

Offset: 0

Vaclav Kotesovec, Feb 27 2019

nonn

Cf. A079478, A324403, A324437, A324438, A324439, A324440, A367834.

Cf. A272246, A307210, A367517, A367543.

a:= n-> mul(mul(i^3+j^3, i=1..n), j=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023

Table[Product[i^3+j^3, {i, 1, n}, {j, 1, n}], {n, 1, 10}]

a(n) = prod(i=1, n, prod(j=1, n, i^3+j^3)); \\ Michel Marcus, Feb 27 2019

from math import prod, factorial
def A324426(n): return prod(i**3+j**3 for i in range(1,n) for j in range(i+1,n+1))**2*factorial(n)**3<Chai Wah Wu, Nov 26 2023

a(n) ~ A * 2^(2*n*(n+1) + 1/4) * exp(Pi*(n*(n+1) + 1/6)/sqrt(3) - 9*n^2/2 - 1/12) * n^(3*n^2 - 3/4) / (3^(5/6) * Pi^(1/6) * Gamma(2/3)^2), where A is the Glaisher-Kinkelin constant A074962.
a(n) = A079478(n) * A367543(n). - Vaclav Kotesovec, Nov 22 2023
For n>0, a(n)/a(n-1) = A272246(n)^2 / (2*n^9). - Vaclav Kotesovec, Dec 02 2023

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

A324438 a(n) = Product_{i=1..n, j=1..n} (i^5 + j^5).

Original entry on oeis.org

1, 2, 139392, 305013568273920000, 1174837791623127613548781790822400000000, 139642003782073074626249921818187528362524804267528306032640000000000000

Offset: 0

Vaclav Kotesovec, Feb 28 2019

nonn

Cf. A079478, A324403, A324426, A324437, A324439, A324440, A367834.

Cf. A272248, A367679.

a:= n-> mul(mul(i^5 + j^5, i=1..n), j=1..n):
seq(a(n), n=0..5);  # Alois P. Heinz, Nov 26 2023

Table[Product[i^5 + j^5, {i, 1, n}, {j, 1, n}], {n, 1, 6}]

from math import prod, factorial
def A324438(n): return prod(i**5+j**5 for i in range(1,n) for j in range(i+1,n+1))**2*factorial(n)**5<Chai Wah Wu, Nov 26 2023

a(n) ~ c * 2^(2*n*(n+1)) * phi^(sqrt(5)*n*(n+1)) * exp(Pi*sqrt(phi)*n*(n+1)/5^(1/4) - 15*n^2/2) * n^(5*n^2 - 5/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio and c = 0.1574073828647726237455544898360432469056972905505624900871695...
a(n) = A367679(n) * A079478(n). - Vaclav Kotesovec, Nov 26 2023
For n>0, a(n)/a(n-1) = A272248(n)^2 / (2*n^15). - Vaclav Kotesovec, Dec 02 2023

a(0)=1 prepended by Alois P. Heinz, Nov 26 2023

A367834 a(n) = Product_{i=1..n, j=1..n} (i^8 + j^8).

Original entry on oeis.org

1, 2, 67634176, 1775927682136440882473213952, 22495149450984565292579847926810488282934424886723006835982336

Offset: 0

Vaclav Kotesovec, Dec 02 2023

nonn

Next term is too long to be included.
For m > 0, Product_{j=1..n, k=1..n} (j^m + k^m) ~ c(m) * exp(n*(n+1)*s(m) - m*n*(n-2)/2) * n^(m*(n^2 - 1/4 - v)), where v = 0 if m > 1 and v = 1/6 if m = 1, s(m) = Sum_{j>=1} (-1)^(j+1)/(j*(1 + m*j)) and c(m) is a constant (dependent only on m). Equivalently, s(m) = log(2) - HurwitzLerchPhi(-1, 1, 1 + 1/m).
c(1) = A / (2^(1/12) * exp(1/12) * sqrt(Pi)).
c(2) = exp(Pi/12) * Gamma(1/4) / (2^(5/4) * Pi^(5/4)).
c(3) = A * 3^(1/6) * exp(Pi/(6*sqrt(3)) - 1/12) * Gamma(1/3)^2 / (2^(7/4) * Pi^(13/6)), where A = A074962 is the Glaisher-Kinkelin constant.
c(4) = A306620.

Cf. A079478 (m=1), A324403 (m=2), A324426 (m=3), A324437 (m=4), A324438 (m=5), A324439 (m=6), A324440 (m=7).

Cf. A306620, A367670, A367833.

Table[Product[i^8 + j^8, {i, 1, n}, {j, 1, n}], {n, 0, 6}]

from math import prod, factorial
def A367834(n): return (prod(i**8+j**8 for i in range(1,n) for j in range(i+1,n+1))*factorial(n)**4)**2<Chai Wah Wu, Dec 02 2023

For n>0, a(n)/a(n-1) = A367833(n)^2 / (2*n^24).

a(n) ~ c * 2^(n*(n+1)) * (1 + 1/(sqrt(1 - 1/sqrt(2)) - 1/2))^(sqrt(2 + sqrt(2))*n*((n+1)/2)) * (1 + 1/(sqrt(1 + 1/sqrt(2)) - 1/2))^(sqrt(2 - sqrt(2))*n*((n+1)/2)) * (n^(8*n^2 - 2) / exp(12*n^2 - Pi*sqrt(1 + 1/sqrt(2))*n*(n+1))), where c = 0.043985703178712025347328240881106818917398444790454628282522057393529338998...

A324439 a(n) = Product_{i=1..n, j=1..n} (i^6 + j^6).

Original entry on oeis.org

1, 2, 1081600, 528465082730906880000, 29276520893554373473343522853366005760000000000, 5719545329208791496596894540018824083491259163047733746620041978183680000000000000000

Offset: 0

Vaclav Kotesovec, Feb 28 2019

nonn

Cf. A079478, A324403, A324426, A324437, A324438, A324440, A367834.

Cf. A367668, A367823.

a:= n-> mul(mul(i^6 + j^6, i=1..n), j=1..n):
seq(a(n), n=0..5);  # Alois P. Heinz, Nov 26 2023

Table[Product[i^6 + j^6, {i, 1, n}, {j, 1, n}], {n, 1, 6}]

from math import prod, factorial
def A324439(n): return (prod(i**6+j**6 for i in range(1,n) for j in range(i+1,n+1))*factorial(n)**3)**2<Chai Wah Wu, Nov 26 2023

a(n) ~ c * 2^(n*(n+1)) * (2 + sqrt(3))^(sqrt(3)*n*(n+1)) * exp(Pi*n*(n+1) - 9*n^2) * n^(6*n^2 - 3/2), where c = 0.104143806044091748191387307161835081649...
a(n) = A324403(n) * A367668(n). - Vaclav Kotesovec, Dec 01 2023
For n>0, a(n)/a(n-1) = A367823^2 / (2*n^18). - Vaclav Kotesovec, Dec 02 2023

a(n)=1 prepended by Alois P. Heinz, Nov 26 2023

A324440 a(n) = Product_{i=1..n, j=1..n} (i^7 + j^7).

Original entry on oeis.org

1, 2, 8520192, 956147263254051187507200, 790929096572487518050439299107158612099228070051840000, 266108022587896795750359251172229660295854509829286134803404773931312693787460334360985600000000000

Offset: 0

Vaclav Kotesovec, Feb 28 2019

nonn

For m>1, Product_{j=1..n, k=1..n} (j^m + k^m) ~ c(m) * exp(n*(n+1)*s(m) - m*n*(n-2)/2) * n^(m*(n^2 - 1/4)), where s(m) = Sum_{j>=1} (-1)^(j+1)/(j*(1 + m*j)) and c(m) is a constant (dependent only on m). Equivalently, s(m) = log(2) - HurwitzLerchPhi(-1, 1, 1 + 1/m). - Vaclav Kotesovec, Dec 01 2023

Cf. A079478, A324403, A324426, A324437, A324438, A324439, A367834.

a:= n-> mul(mul(i^7 + j^7, i=1..n), j=1..n):
seq(a(n), n=0..5);  # Alois P. Heinz, Nov 26 2023

Table[Product[i^7+j^7, {i, 1, n}, {j, 1, n}], {n, 1, 6}]

from math import prod, factorial
def A324440(n): return prod(i**7+j**7 for i in range(1,n) for j in range(i+1,n+1))**2*factorial(n)**7<Chai Wah Wu, Nov 26 2023

Limit_{n->oo} (a(n)^(1/n^2))/n^7 = 2^(3/2) * (cos(3*Pi/14) / tan(Pi/7))^sin(3*Pi/14) / ((cos(Pi/14)*tan(3*Pi/14))^sin(Pi/14) * (sin(Pi/7)*tan(Pi/14))^cos(Pi/7)) * exp((Pi/sin(Pi/7) - 21)/2) = 0.0334234967249533921390751418772468470887965377...
From Vaclav Kotesovec, Dec 01 2023: (Start)
a(n) ~ c * exp(n*(n+1)*s - 7*n*(n-2)/2) * n^(7*(n^2 - 1/4)), where
s = Sum_{j>=1} (-1)^(j+1)/(j*(1 + 7*j)) = Pi/(2*sin(Pi/7)) + 3*log(2)/2 - 7 - cos(Pi/7) * log(2*sin(Pi/14)^2) - log(2*sin(3*Pi/14)^2) * sin(Pi/14) + log(cos(3*Pi/14)*cos(Pi/7) / sin(Pi/7)) * sin(3*Pi/14) = 0.10150386842315637912206687298894641634315636548242136512503... and
c = 0.068056503846689328929612652207251071282623125565150941566636264805878144...
Equivalently, s = log(2) - HurwitzLerchPhi(-1, 1, 1 + 1/7). (End)

a(0)=1 prepended by Alois P. Heinz, Nov 26 2023

A367668 a(n) = Product_{i=1..n, j=1..n} (i^4 - i^2*j^2 + j^4).

Original entry on oeis.org

1, 2704, 4343072672016, 104066856161782811235776987136, 368057974579278182597141600363036562863943425064960000, 1139317987311004502889916180807286481186277543437822119282797720728081762451885916160000

Offset: 1

Vaclav Kotesovec, Nov 26 2023

nonn

Cf. A203675, A324437, A367550.

Table[Product[Product[i^4 - i^2*j^2 + j^4, {i, 1, n}], {j, 1, n}], {n, 1, 10}]

from math import prod, factorial
def A367668(n): return (prod((k:=j**2)**2+(m:=i**2)*(m-k) for i in range(1,n) for j in range(i+1,n+1))*factorial(n)**2)**2 # Chai Wah Wu, Nov 26 2023

a(n) ~ c * (2 + sqrt(3))^(sqrt(3)*n*(n+1)) * n^(4*n^2 - 1) / exp(6*n^2 - Pi*n*(n+1)/2), where c = 0.219927317102868518491484945565471919409874745762951216457178735860943437...

A367550 a(n) = Product_{i=1..n, j=1..n} (i^4 + i^2*j^2 + j^4).

Original entry on oeis.org

3, 63504, 2260442279270448, 3379470372507391964272022793486336, 2097229364987262298214192667129919538956418868293588090880000

Offset: 1

Vaclav Kotesovec, Nov 22 2023

nonn

Cf. A203673, A324437, A367542, A367543, A367668.

Table[Product[Product[i^4 + i^2*j^2 + j^4, {i, 1, n}], {j, 1, n}], {n, 1, 10}]

from math import prod, factorial
def A367550(n): return (prod((i2:=i**2)*(i2+(j2:=j**2))+j2**2 for i in range(1,n) for j in range(i+1,n+1))*factorial(n)**2)**2*3**n # Chai Wah Wu, Nov 22 2023

a(n) = A367542(n) * A367543(n).

a(n) ~ Gamma(1/3)^3 * 3^(3*n*(n+1)/2 + 7/12) * n^(4*n^2 - 1) / (8*Pi^3 * exp(6*n^2 - (6*n*(n+1) + 1)*Pi/(4*sqrt(3)))).

A203677 Vandermonde sequence using x^2 + y^2 applied to (1,4,9,...,n^2).

Original entry on oeis.org

1, 17, 135218, 3185418047264, 795022479172023183220864, 5554004683279652358469137440150614769664, 2378852972988348412358457063032448409092378064835941488918528

Offset: 1

Clark Kimberling, Jan 04 2012

nonn

See A093883 for a discussion and guide to related sequences.

Cf. A324437.

f[j_] := j^2; z = 12;
u[n_] := Product[f[j]^2 + f[k]^2, {j, 1, k - 1}]
v[n_] := Product[u[n], {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203677 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203678 *)

a(n) ~ c * 2^(n^2/2 - 1) * (1 + sqrt(2))^(n*(n+1)/sqrt(2)) * exp((Pi/2^(3/2) - 3)*n^2 + (Pi/2^(3/2) + 2)*n) * n^(2*n^2 - 2*n - 3/2), where c = 0.154147406559582639039828423669556073435424655001221440918550218582474208... - Vaclav Kotesovec, Sep 08 2023

cp's OEIS Frontend

A306620 Decimal expansion of a constant related to the asymptotics of A324437.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A324426 a(n) = Product_{i=1..n, j=1..n} (i^3 + j^3).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A324438 a(n) = Product_{i=1..n, j=1..n} (i^5 + j^5).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Python

Formula

Extensions

A367834 a(n) = Product_{i=1..n, j=1..n} (i^8 + j^8).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Python

Formula

A324439 a(n) = Product_{i=1..n, j=1..n} (i^6 + j^6).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Python

Formula

Extensions

A324440 a(n) = Product_{i=1..n, j=1..n} (i^7 + j^7).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

Python

Formula

Extensions

A367668 a(n) = Product_{i=1..n, j=1..n} (i^4 - i^2*j^2 + j^4).

Views