A367958 -id:A367958 - cp's OEIS frontend

A324402 a(n) = Product_{i=1..n, j=1..n} (2*i + j).

1, 3, 360, 6350400, 36212520960000, 117563342374788710400000, 337905477880065368190647009280000000, 1234818479230749311108497004714406224855040000000000, 7795494015765035913020359514023640290443493305037073940480000000000000

Offset: 0

Vaclav Kotesovec, Feb 26 2019

nonn

Robert Israel, Table of n, a(n) for n = 0..25

Cf. A059486, A079478, A324403, A367956, A367957, A367958.

f:= n -> mul((2*i+n)!/(2*i)!,i=1..n):
map(f, [$0..10]); # Robert Israel, Feb 27 2019

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

a(n) ~ sqrt(A/Pi) * 3^(9*n*(n+1)/4 + 11/24) * n^(n^2 - 11/24) / (2^(n^2 + 3*n/2 + 17/24) * exp(3*n^2/2 + 1/24)), where A is the Glaisher-Kinkelin constant A074962.
a(n) = 3*n*a(n-1)*Product_{i=1..n-1} (2*i+n)(2*n+i). - Chai Wah Wu, Feb 26 2019
a(n) = a(n-1) * (3*n)! * (3*n-2)!!/((2*n)! * n!!). - Robert Israel, Feb 27 2019

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

A367944 a(n) = Product_{i=1..n, j=1..n} (i^2 + 5*j^2).

Original entry on oeis.org

1, 6, 27216, 1344924798336, 3605580335899213007486976, 1648055031941075082958467426002632704000000, 312704667066499295437237787452750428210311485710262201221120000000

Offset: 0

Vaclav Kotesovec, Dec 05 2023

nonn

In general, for d>0, Product_{i=1..n, j=1..n} (i^2 + d*j^2) ~ c(d) * n^(2*n^2 - 1/2) * (d+1)^(n*(n+1)) * d^(-n/2) * exp(n*(n+1)*(Pi*d/2 - (d-1)*arctan(sqrt(d))) / sqrt(d) - 3*n^2), where c(d) is a constant (dependent only on d).

c(1) = exp(Pi/12) * Gamma(1/4) / (2*Pi)^(5/4), cf. A324403.

Cf. A324403 (d=1), A367941 (d=2), A367942 (d=3), A367943 (d=4).

Cf. A367542, A367543, A367958.

Table[Product[i^2+5*j^2, {i, 1, n}, {j, 1, n}], {n, 0, 8}]

a(n) ~ c * n^(2*n^2 - 1/2) * 6^(n*(n+1)) * 5^(-n/2) * exp(n*(n+1)*(5*Pi/2 - 4*arctan(sqrt(5)))/sqrt(5) - 3*n^2), where c = 0.4431081869167792949266065295798218232844989957987096447783995373751372668...

A367956 a(n) = Product_{i=1..n, j=1..n} (i + 3*j).

Original entry on oeis.org

1, 4, 1120, 79833600, 3173289799680000, 123650071173117090201600000, 7337799401269093351612002462597120000000, 951792703318385182295191545713146608287219712000000000000

Offset: 0

Vaclav Kotesovec, Dec 06 2023

nonn

Cf. A074962, A079478, A324402, A367957, A367958.

Table[Product[i + 3*j, {i, 1, n}, {j, 1, n}], {n, 0, 10}]

a(n) ~ A^(1/3) * 2^(16*n*(n+1)/3 + 13/18) * n^(n^2 - 19/36) / (Pi^(1/3) * Gamma(1/3)^(1/3) * 3^(n*(3*n+4)/2 + 11/36) * exp(3*n^2/2 + 1/36)), where A = A074962 is the Glaisher-Kinkelin constant.

A367957 a(n) = Product_{i=1..n, j=1..n} (i + 4*j).

Original entry on oeis.org

1, 5, 2700, 567567000, 101370917007360000, 26995322179162164731904000000, 16635639072295355604762223305031680000000000, 34026881962001914598329145027742925521204742717440000000000000

Offset: 0

Vaclav Kotesovec, Dec 06 2023

nonn

Cf. A074962, A079478, A324402, A367956, A367958.

Table[Product[i + 4*j, {i, 1, n}, {j, 1, n}], {n, 0, 10}]

a(n) ~ A^(1/4) * 5^(25*n*(n+1)/8 + 29/48) * n^(n^2 - 29/48) / (Pi^(1/4) * Gamma(1/4)^(1/2) * 2^(n*(4*n+5) + 5/6) * exp(3*n^2/2 + 1/48)), where A = A074962 is the Glaisher-Kinkelin constant.

cp's OEIS Frontend

A324402 a(n) = Product_{i=1..n, j=1..n} (2*i + j).

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A367944 a(n) = Product_{i=1..n, j=1..n} (i^2 + 5*j^2).

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A367956 a(n) = Product_{i=1..n, j=1..n} (i + 3*j).

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A367957 a(n) = Product_{i=1..n, j=1..n} (i + 4*j).

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