Displaying 1-10 of 33 results found.
a(n) = Product_{i=1..n, j=1..n} (i^4 + j^4).
+10
15
1, 2, 18496, 189567553208832, 53863903330477722171391434817536, 4194051697335929481600368256016484482740174637152337920000, 530545265060440849231458462212366841894726534233233018777709463062563850450708386692464640000
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...
MAPLE
a:= n-> mul(mul(i^4+j^4, i=1..n), j=1..n):
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
a(n) = Product_{i=1..n, j=1..n} (i^3 + j^3).
+10
14
1, 2, 2592, 134425267200, 3120795915109442519040000, 180825857777547616919759624941086965760000000, 99356698720512072045648926659510730227553351200000695922065408000000000
FORMULA
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.
MAPLE
a:= n-> mul(mul(i^3+j^3, i=1..n), j=1..n):
MATHEMATICA
Table[Product[i^3+j^3, {i, 1, n}, {j, 1, n}], {n, 1, 10}]
PROG
(PARI) a(n) = prod(i=1, n, prod(j=1, n, i^3+j^3)); \\ Michel Marcus, Feb 27 2019 (Python)
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<<n # Chai Wah Wu, Nov 26 2023
a(n) = Product_{i=1..n, j=1..n} (i^2 - i*j + j^2).
+10
14
1, 36, 777924, 51190934086656, 32435802373365731229926400, 483207398728525904876601066508152707481600, 350969035472356907726779584093506665415605824531908346799718400
FORMULA
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
a(n) = Product_{i=1..n, j=1..n} (i^2 + i*j + j^2).
+10
13
3, 1764, 2905736652, 66016970246853190656, 64657853715047202043531429875379200, 6627957368676918780503749855130249245999452089509478400
FORMULA
a(n) ~ c * 3^(3*n*(n+1)/2) * n^(2*n^2 - 2/3) / exp(3*n^2 - Pi*n*(n+1) / (2*sqrt(3))), where c = 3^(5/12) * exp(Pi/(12*sqrt(3))) * Gamma(1/3) / (2^(4/3) * Pi^(4/3)) = 0.42478290981890921418850643030484274341562970375995548434917...
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 A367542(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*3**n # Chai Wah Wu, Nov 22 2023
a(n) = Product_{k=0..n} (n^2 + k^2).
+10
12
0, 2, 160, 21060, 4352000, 1313845000, 547573478400, 301758856490000, 212663770808320000, 186659516597629140000, 199722414913149440000000, 255947740845844788169000000, 387074162712817024892928000000, 682170272459193898736228210000000
FORMULA
a(n) ~ 2^(n + 1/2) * n^(2*(n+1)) / exp((4-Pi)*n/2).
MATHEMATICA
Table[Product[n^2+k^2, {k, 0, n}], {n, 0, 15}]
a(n) = Product_{i=1..n, j=1..n, k=1..n} (i^2 + j^2 + k^2).
+10
9
1, 3, 5668704, 550388591715704109656479285248, 152455602303300418998634460043817052571893573096619261814850281699755319515987050496
COMMENTS
(a(n)^(1/n^3))/n^2 tends to 0.828859579669279... = A306617.
MAPLE
a:= n-> mul(mul(mul(i^2+j^2+k^2, i=1..n), j=1..n), k=1..n):
MATHEMATICA
Table[Product[i^2+j^2+k^2, {i, 1, n}, {j, 1, n}, {k, 1, n}], {n, 1, 6}]
Clear[a]; a[n_] := a[n] = If[n == 1, 3, a[n-1] * Product[k^2 + j^2 + n^2, {j, 1, n}, {k, 1, n}]^3 * (3*n^2) / (Product[k^2 + 2*n^2, {k, 1, n}]^3)]; Table[a[n], {n, 1, 6}] (* Vaclav Kotesovec, Mar 27 2019 *)
a(n) = Product_{i=1..n, j=1..n} (i^5 + j^5).
+10
9
1, 2, 139392, 305013568273920000, 1174837791623127613548781790822400000000, 139642003782073074626249921818187528362524804267528306032640000000000000
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...
MAPLE
a:= n-> mul(mul(i^5 + j^5, i=1..n), j=1..n):
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
a(n) = Product_{i=1..n, j=1..n} (i^8 + j^8).
+10
9
1, 2, 67634176, 1775927682136440882473213952, 22495149450984565292579847926810488282934424886723006835982336
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.
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
a(n) = Product_{i=1..n, j=1..n} (i^6 + j^6).
+10
8
1, 2, 1081600, 528465082730906880000, 29276520893554373473343522853366005760000000000, 5719545329208791496596894540018824083491259163047733746620041978183680000000000000000
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...
MAPLE
a:= n-> mul(mul(i^6 + j^6, i=1..n), j=1..n):
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
a(n) = Product_{1 <= i < j <= n} (i^2 + j^2).
+10
7
1, 5, 650, 5525000, 5807194900000, 1226800120038480000000, 77092420109247492627600000000000, 2001314057760220784660590245696000000000000000, 28468550112906756205383102673584071297339520000000000000000000
COMMENTS
Each term divides its successor, as in A203476.
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):
MATHEMATICA
f[j_]:= j^2; z = 15;
v[n_]:= Product[Product[f[k] + f[j], {j, k-1}], {k, 2, n}]
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
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