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

Author

Vaclav Kotesovec, Feb 28 2019

Keywords

• nonn

Comments

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

Crossrefs

Cf. A079478, A306594.

Programs

Maple

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

Mathematica

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

Formula

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...

Extensions

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