A367956 -id:A367956 - 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

A367958 a(n) = Product_{i=1..n, j=1..n} (i + 5*j).

Original entry on oeis.org

1, 6, 5544, 2822916096, 1723467782592331776, 2210440498434925488635904000000, 9234659938893939743399592700454853672960000000, 180150216814109052335771891722360520401032374209013927116800000000

Offset: 0

Vaclav Kotesovec, Dec 06 2023

nonn

In general, for d>0, Product_{i=1..n, j=1..n} (i + d*j) ~ A^(1/d) * (Product_{j=1..d} Gamma(j/d)^(j/d)) * (d+1)^((d/2 + 1 + 1/(2*d))*n*(n+1) + (d+1)^2/(12*d) + 1/12) * n^(n^2 - d/12 - 1/4 - 1/(12*d)) / ((2*Pi)^((d+1)/4) * exp(3*n^2/2 + 1/(12*d)) * d^((n*(d*n + (d+1)))/2 - 1/(12*d))), where A = A074962 is the Glaisher-Kinkelin constant.

Equivalently, for d>0, Product_{i=1..n, j=1..n} (i + d*j) ~ A^d * (Product_{j=1..d} BarnesG(j/d)) * (2*Pi)^((d-3)/4) * (d+1)^((d + (d+1)^2*(6*n*(n+1) + 1)) / (12*d)) * n^(n^2 - 1/4 - 1/(12*d) - d/12) / (d^((n+1)*(d*n + 1)/2) * exp(3*n^2/2 + d/12)).

Cf. A079478 (d=1), A324402 (d=2), A367956 (d=3), A367957 (d=4).

Cf. A074962, A367842, A367944.

a:= n-> mul(mul(i+5*j, i=1..n), j=1..n):
seq(a(n), n=0..8);  # Alois P. Heinz, Dec 06 2023

Table[Product[i + 5*j, {i, 1, n}, {j, 1, n}], {n, 0, 10}]

a(n) ~ A^(1/5) * (1 + sqrt(5))^(1/10) * 2^(18*n*(n+1)/5 + 29/60) * 3^(18*n*(n+1)/5 + 41/60) * n^(n^2 - 41/60) / (Pi^(1/10) * Gamma(1/5)^(3/5) * Gamma(2/5)^(1/5) * 5^(n*(5*n+6)/2 + 1/3) * exp(3*n^2/2 + 1/60)), 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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A367958 a(n) = Product_{i=1..n, j=1..n} (i + 5*j).

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

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