A324426 -id:A324426

A324403

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

+10
34

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

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

OFFSET

0,2

COMMENTS

Next term is too long to be included.

LINKS

Table of n, a(n) for n=0..7.

FORMULA

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

MAPLE

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

MATHEMATICA

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

PROG

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

(Python)

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<<n # Chai Wah Wu, Nov 22 2023

CROSSREFS

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

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

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Feb 26 2019

EXTENSIONS

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

STATUS

approved

A324437

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

+10
15

1, 2, 18496, 189567553208832, 53863903330477722171391434817536, 4194051697335929481600368256016484482740174637152337920000, 530545265060440849231458462212366841894726534233233018777709463062563850450708386692464640000

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

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..6.

FORMULA

a(n) ~ c * 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), where c = A306620 = 0.23451584451404279281807143317500518660696293944961...

For n>0, a(n)/a(n-1) = A272247(n)^2 / (2*n^12). - Vaclav Kotesovec, Dec 01 2023

MAPLE

a:= n-> mul(mul(i^4+j^4, i=1..n), j=1..n):

seq(a(n), n=0..7); # Alois P. Heinz, Jun 24 2023

MATHEMATICA

Table[Product[i^4 + j^4, {i, 1, n}, {j, 1, n}], {n, 1, 6}]

PROG

(Python)

from math import prod, factorial

def A324437(n): return (prod(i**4+j**4 for i in range(1, n) for j in range(i+1, n+1))*factorial(n)**2)**2<<n # Chai Wah Wu, Nov 26 2023

CROSSREFS

Cf. A079478, A324403, A324426, A324438, A324439, A324440, A367834.

Cf. A203677, A272247, A307215.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Feb 28 2019

EXTENSIONS

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

STATUS

approved

A367543

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

+10
14

1, 36, 777924, 51190934086656, 32435802373365731229926400, 483207398728525904876601066508152707481600, 350969035472356907726779584093506665415605824531908346799718400

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

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..7.

FORMULA

a(n) = A324426(n) / A079478(n).

a(n) ~ 3^(1/6) * Gamma(1/3)^2 * n^(2*n^2 - 1/3) / (2^(5/3) * Pi^(5/3) * exp(3*n^2 - (n^2 + n + 1/6)*Pi/sqrt(3))).

MATHEMATICA

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

PROG

(Python)

from math import prod, factorial

def A367543(n): return (prod(i*(i-j)+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

CROSSREFS

Cf. A203312, A324403, A367542.

Cf. A324426, A079478.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Nov 22 2023

STATUS

approved

A324438

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

+10
9

1, 2, 139392, 305013568273920000, 1174837791623127613548781790822400000000, 139642003782073074626249921818187528362524804267528306032640000000000000

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

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..5.

FORMULA

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

MAPLE

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

MATHEMATICA

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

PROG

(Python)

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<<n # Chai Wah Wu, Nov 26 2023

CROSSREFS

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

Cf. A272248, A367679.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Feb 28 2019

EXTENSIONS

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

STATUS

approved

A367834

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

+10
9

1, 2, 67634176, 1775927682136440882473213952, 22495149450984565292579847926810488282934424886723006835982336

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

OFFSET

0,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..4.

FORMULA

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...

MATHEMATICA

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

PROG

(Python)

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<<n # Chai Wah Wu, Dec 02 2023

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Dec 02 2023

STATUS

approved

A324439

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

+10
8

1, 2, 1081600, 528465082730906880000, 29276520893554373473343522853366005760000000000, 5719545329208791496596894540018824083491259163047733746620041978183680000000000000000

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

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..5.

FORMULA

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

MAPLE

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

MATHEMATICA

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

PROG

(Python)

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<<n # Chai Wah Wu, Nov 26 2023

CROSSREFS

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

Cf. A367668, A367823.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Feb 28 2019

EXTENSIONS

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

STATUS

approved

A272246

a(n) = Product_{k=0..n} (n^3 + k^3).

+10
7

0, 2, 1152, 1428840, 3488808960, 15044494500000, 105235903511101440, 1119277024472896248960, 17216259547948971039129600, 368066786222106315186876633600, 10591209807103301277597696000000000, 399472472359100444604916002033020774400

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

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..11.

FORMULA

a(n) ~ 2^(2*n + 1/2) * n^(3*n + 3) / exp((3 - Pi/sqrt(3))*n).

MATHEMATICA

Table[Product[n^3+k^3, {k, 0, n}], {n, 0, 12}]

CROSSREFS

Cf. A126804, A272244, A272247, A272248, A367823, A367833.

Cf. A255433, A323541, A324426.

KEYWORD

nonn,easy

AUTHOR

Vaclav Kotesovec, Apr 23 2016

STATUS

approved

A324440

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

+10
7

1, 2, 8520192, 956147263254051187507200, 790929096572487518050439299107158612099228070051840000, 266108022587896795750359251172229660295854509829286134803404773931312693787460334360985600000000000

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

OFFSET

0,2

COMMENTS

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

LINKS

Table of n, a(n) for n=0..5.

FORMULA

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)

MAPLE

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

MATHEMATICA

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

PROG

(Python)

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<<n # Chai Wah Wu, Nov 26 2023

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Feb 28 2019

EXTENSIONS

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

STATUS

approved

A368722

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

+10
4

1, 3, 353736000, 4795587079853800623303366670123008000000

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

OFFSET

0,2

COMMENTS

Next term is too long to be included.

LINKS

Table of n, a(n) for n=0..3.

FORMULA

Limit_{n->oo} a(n)^(1/(n^3)) / n^3 = exp(Integral_{x=0..1, y=0..1, z=0..1} log(x^3 + y^3 + z^3) dz dy dx) = 0.5334736919092639993380174031245...

MATHEMATICA

Table[Product[i^3 + j^3 + k^3, {i, 1, n}, {j, 1, n}, {k, 1, n}], {n, 0, 5}]

CROSSREFS

Cf. A306594, A324425, A368723.

Cf. A324426, A368720.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Jan 04 2024

STATUS

approved

A307209

Decimal expansion of Product_{i>=1, j>=1} (1 + 1/(i^3 + j^3)).

+10
3

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

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

OFFSET

1,1

COMMENTS

Product_{i>=1, j>=1} (1 + 1/(i^2 + j^2)) is divergent.

A324443(n) / A324403(n) ~ c * n^(Pi/2), where c = A306398 * 2^(3/4) * exp(-Pi/12) * Pi^(1/4) * Gamma(3/4) = 0.36753062884677326134620846786416595535234038999313...

Product_{i=1..n, j=1..n} (1 + 1/(i + j)) = A324444(n) / A079478(n) ~ 2^(2*n + 1) / (sqrt(Pi)*n^(3/2)).

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..224

FORMULA

Equals limit_{n->infinity} A307210(n) / A324426(n).

EXAMPLE

3.50478299933972837589112057043806125583893247862712753541994626614058385...

MATHEMATICA

(* The iteration cycle: *) $MaxExtraPrecision = 1000; funs[n_] := Product[1 + 1/(i^3 + j^3), {i, 1, n}, {j, 1, n}]; Do[Print[N[Sum[(-1)^(m + j)*funs[j*Floor[200/m]] * j^(m - 1)/(j - 1)!/(m - j)!, {j, 1, m}], 100]], {m, 10, 100, 10}]

PROG

(PARI) default(realprecision, 50); exp(sumalt(k=1, -(-1)^k/k*sumnum(i=1, sumnum(j=1, 1/(i^3+j^3)^k)))) \\ 15 decimals correct

CROSSREFS

Cf. A073017, A307210, A307215, A324403, A324426, A324443.

KEYWORD

nonn,cons

AUTHOR

Vaclav Kotesovec, Mar 28 2019

STATUS

approved