A079478 -id:A079478 - cp's OEIS frontend

A324441 a(n) = Product_{k1=1..n, k2=1..n, k3=1..n, k4=1..n} (k1 + k2 + k3 + k4).

1, 4, 2240421120000, 2357018782335863659143506877669927151046989269393693317529600000000000000

Offset: 0

Vaclav Kotesovec, Feb 28 2019

nonn

Next term is too long to be included.
Limit_{n->oo} ((Product_{k1=1..n, k2=1..n, k3=1..n, k4=1..n, k5=1..n} (k1 + k2 + k3 + k4 + k5))^(1/n^5))/n = 2^(-88) * 3^(81/4) * 5^(625/24) * exp(-137/60).
Limit_{n->oo} ((Product_{k1=1..n, k2=1..n, k3=1..n, k4=1..n, k5=1..n, k6=1..n} (k1 + k2 + k3 + k4 + k5 + k6))^(1/n^6))/n = 2^(1184/5) * 3^(891/20) * 5^(-3125/24) * exp(-49/20).
Limit_{n->oo} ((Product_{k1=1..n, k2=1..n, k3=1..n, k4=1..n, k5=1..n, k6=1..n, k7=1..n} (k1 + k2 + k3 + k4 + k5 + k6 + k7))^(1/n^7))/n = 2^(-5552/9) * 3^(-29889/80) * 5^(15625/48) * 7^(117649/720) * exp(-363/140).
From Vaclav Kotesovec, Dec 23 2023: (Start)
Limit_{n->oo} ((Product_{k1=1..n, k2=1..n, k3=1..n, k4=1..n, k5=1..n, k6=1..n, k7=1..n, k8=1..n} (k1 + k2 + k3 + k4 + k5 + k6 + k7 + k8))^(1/n^8))/n = 2^(277456/105) * 3^(92583/80) * 5^(-78125/144) * 7^(-823543/720) * exp(-761/280).
Limit_{n->oo} ((Product_{k1=1..n, k2=1..n, k3=1..n, k4=1..n, k5=1..n, k6=1..n, k7=1..n, k8=1..n, k9=1..n} (k1 + k2 + k3 + k4 + k5 + k6 + k7 + k8 + k9))^(1/n^9))/n = 2^(-37504/3) * 3^(-432297/2240) * 5^(390625/576) * 7^(5764801/1440) * exp(-7129/2520). (End)
In general, for m >= 1, limit_{n->oo} ((Product_{k1=1..n, k2=1..n, ... , km=1..n} (k1 + k2 + ... + km))^(1/n^m))/n = exp(-HarmonicNumber(m)) * Product_{j=1..m} j^((-1)^(m-j) * j^m / (j! * (m-j)!)). - Vaclav Kotesovec, Dec 26 2023

Cf. A079478, A306594.

a:= n-> mul(mul(mul(mul(i+j+k+m, i=1..n), j=1..n), k=1..n), m=1..n):
seq(a(n), n=0..4);  # Alois P. Heinz, Jun 24 2023

Table[Product[k1 + k2 + k3 + k4, {k1, 1, n}, {k2, 1, n}, {k3, 1, n}, {k4, 1, n}], {n, 1, 5}]

Limit_{n->oo} (a(n)^(1/n^4))/n = 2^(76/3) * 3^(-27/2) * exp(-25/12) = exp(Integral_{k1=0..1, k2=0..1, k3=0..1, k4=0..1} log(k1 + k2 + k3 + k4) dk4 dk3 dk2 dk1) = 1.9062335728830251698721203...

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

A367679 a(n) = Product_{i=1..n, j=1..n} (i^4 - i^3j + i^2j^2 - i*j^3 + j^4).

Original entry on oeis.org

1, 1936, 1765124816400, 19271059559619728900751360000, 25048411180596698786915756280274804766474649600000000, 23045227505577134384745253646275782332295626096040088365089618773238077194240000000000

Offset: 1

Vaclav Kotesovec, Nov 26 2023

nonn

Cf. A079478, A324437, A324438, A367550, A367668.

Table[Product[i^4 - i^3*j + i^2*j^2 - i*j^3 + j^4, {i, 1, n}, {j, 1, n}], {n, 1, 6}]

from math import prod, factorial
def A367679(n): return (prod(i*(i*(i*(i-j)+j**2)-j**3)+j**4 for i in range(1,n) for j in range(i+1,n+1))*factorial(n)**2)**2 # Chai Wah Wu, Nov 26 2023

a(n) = A324438(n) / A079478(n).

a(n) ~ c * n^(4*n^2 - 5/6) * phi^(sqrt(5)*n*(n+1)) / exp(6*n^2 - sqrt(phi)*Pi*n*(n+1)/5^(1/4)), where phi = A001622 is the golden ratio and c = 0.2505211390193028244009922677012518708897316924498037078191143761182342931773594...

A107251 Supercatalan numbers SF(2n)/(SF(n)*SF(n+1)) where SF is the superfactorial function A000178.

Original entry on oeis.org

1, 1, 12, 7200, 508032000, 7742895390720000, 40797452088662556672000000, 108985983996792124183843071590400000000, 203800994173724454677862841368011757060096000000000000

Offset: 0

Henry Bottomley, May 14 2005

nonn

			a(3) = 1!*2!*3!*4!*5!*6!/(1!*2!*3!*1!*2!*3!*4!) = 24883200/(12*288) = 7200.

Cf. A000108 for original Catalan numbers (2n)!/(n!*(n+1)!).

Cf. A000142, A000178, A079478.

seq(mul(mul(k+j,j=1..n), k=2..n), n=0..8); # Zerinvary Lajos, Jun 01 2007

a(n) = (n+2)!*(n+3)!*...*(2n)!/(2!*3!*...*n!) = A000178(2n)/(A000178(n)*A000178(n+1)) = A079478(n)/A000142(n+1).

a(n) ~ A * 2^(2*n^2 + 2*n - 7/12) * n^(n^2 - n - 23/12) / (Pi * exp(3*n^2/2 - n + 1/12)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 10 2015

A368686 a(n) = Product_{j=0..n, k=0..n} (j + k + n).

Original entry on oeis.org

0, 12, 172800, 1536288768000, 16189465114548633600000, 322110526445545505917029580800000000, 17555281051920416386104936570114748195012608000000000, 3580285185706909590176164870311607533516764550107699116769280000000000000

Offset: 0

Vaclav Kotesovec, Jan 03 2024

nonn

Cf. A079478, A306594, A324444, A368622, A368685.

Table[Product[i+j+n, {i, 0, n}, {j, 0, n}], {n, 0, 8}]
Join[{0}, Table[3*n*BarnesG[n] * BarnesG[3*n] * Gamma[3*n]^2 / BarnesG[2*n+1]^2, {n, 1, 8}]]

For n>0, a(n) = 3*n*BarnesG(n) * BarnesG(3*n) * Gamma(3*n)^2 / BarnesG(2*n+1)^2.
a(n) ~ 3^(9*n^2/2 + 3*n + 5/12) * n^((n+1)^2) / (2^(4*n^2 - 1/6) * exp(3*n^2/2 + 2*n)).
a(n) = 4*n*Gamma(2*n)^2 * A368685(n) / Gamma(n)^2.

A324442 a(n) = Product_{i=1..n, j=1..n} (i^2 + j).

Original entry on oeis.org

1, 2, 180, 6652800, 402265543680000, 109211487076824381849600000, 295382703175843424854047228769075200000000, 15385012566245626089929288743828190926813939944652800000000000

Offset: 0

Vaclav Kotesovec, Feb 28 2019

nonn

Cf. A079478, A101686, A255322, A324403.

a:= n-> mul(mul(i^2+j, i=1..n), j=1..n):
seq(a(n), n=0..8);  # Alois P. Heinz, Jun 24 2023

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

From Vaclav Kotesovec, Dec 27 2023: (Start)
a(n) ~ c * n^(2*n^2 + n/2 - 1/4) / exp(2*n^2 - 2*Pi*n^(3/2)/3 - Pi*sqrt(n)/2), where c = 0.31906...
For n>1, a(n) = a(n-1) * Gamma(n - i*sqrt(n)) * Gamma(n + i*sqrt(n)) * Gamma(n^2 + n + 1) * sinh(Pi*sqrt(n)) / (Pi * n^(5/2) * Gamma(n^2)), where i is the imaginary unit. (End)

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

cp's OEIS Frontend

A324441 a(n) = Product_{k1=1..n, k2=1..n, k3=1..n, k4=1..n} (k1 + k2 + k3 + k4).

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A367679 a(n) = Product_{i=1..n, j=1..n} (i^4 - i^3*j + i^2*j^2 - i*j^3 + j^4).

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A107251 Supercatalan numbers SF(2n)/(SF(n)*SF(n+1)) where SF is the superfactorial function A000178.

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A368686 a(n) = Product_{j=0..n, k=0..n} (j + k + n).

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A324442 a(n) = Product_{i=1..n, j=1..n} (i^2 + j).

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A367679 a(n) = Product_{i=1..n, j=1..n} (i^4 - i^3j + i^2j^2 - i*j^3 + j^4).