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

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

Offset

0,2

Comments

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)).

Links

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

Formula

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.

Maple

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

Mathematica

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

Crossrefs

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

Cf. A074962, A367842, A367944.

Sequence in context: A115431 A116117 A116135 * A011788 A320446 A172649

Adjacent sequences: A367955 A367956 A367957 * A367959 A367960 A367961

Keyword

nonn

Author

Vaclav Kotesovec, Dec 06 2023

Status

approved