cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

A324425 a(n) = Product_{i=1..n, j=1..n, k=1..n} (i^2 + j^2 + k^2).

Original entry on oeis.org

1, 3, 5668704, 550388591715704109656479285248, 152455602303300418998634460043817052571893573096619261814850281699755319515987050496
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 27 2019

Keywords

Comments

(a(n)^(1/n^3))/n^2 tends to 0.828859579669279... = A306617.

Crossrefs

Cf. A079478, A306594, A368722, A368723.
Cf. A324403, A306617, A368622, A368623.

Programs

  • Maple
    a:= n-> mul(mul(mul(i^2+j^2+k^2, i=1..n), j=1..n), k=1..n):
seq(a(n), n=0..5);  # Alois P. Heinz, Jun 24 2023
  • Mathematica
    Table[Product[i^2+j^2+k^2, {i, 1, n}, {j, 1, n}, {k, 1, n}], {n, 1, 6}]
Clear[a]; a[n_] := a[n] = If[n == 1, 3, a[n-1] * Product[k^2 + j^2 + n^2, {j, 1, n}, {k, 1, n}]^3 * (3*n^2) / (Product[k^2 + 2*n^2, {k, 1, n}]^3)]; Table[a[n], {n, 1, 6}] (* Vaclav Kotesovec, Mar 27 2019 *)

Extensions

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

A324444 a(n) = Product_{i=1..n, j=1..n} (1 + i + j).

Original entry on oeis.org

1, 3, 240, 1512000, 1536288768000, 429266120461516800000, 50406068004584362019389440000000, 3534677027377560888380072035048488960000000000, 199761495428405897006583857561824669625759249203200000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 28 2019

Keywords

Crossrefs

Cf. A079478, A306594.

Programs

  • Magma
    [(&*[(&*[1+k+j: j in [1..n]]): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Feb 28 2019
  • Maple
    a:= n-> mul(mul(i+j+1, i=1..n), j=1..n):
seq(a(n), n=0..9);  # Alois P. Heinz, Jun 24 2023
  • Mathematica
    Table[Product[1 + i + j, {i, 1, n}, {j, 1, n}], {n, 1, 10}]
Table[BarnesG[2*n + 3] / BarnesG[n + 3]^2, {n, 1, 10}]
  • PARI
    a(n) = prod(i=1, n, prod(j=1, n, 1+i+j)); \\ Michel Marcus, Feb 28 2019
  • Sage
    [product( product(1+k+j for j in (1..n)) for k in (1..n)) for n in (1..10)] # G. C. Greubel, Feb 28 2019

Formula

a(n) ~ A * 2^(2*n^2 + 4*n + 11/12) * n^(n^2 - 23/12) / (Pi * exp(3*n^2/2 + 1/12)), where A is the Glaisher-Kinkelin constant A074962.
a(n) = BarnesG(2*n + 3) / BarnesG(n + 3)^2.
Product_{i=1..n, j=1..n} (1 + 1/(i + j)) = a(n) / A079478(n) ~ 2^(2*n + 1) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 29 2019

Extensions

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

A093884 Product of all possible sums of three numbers taken from among first n natural numbers.

Original entry on oeis.org

6, 3024, 2874009600, 159950125679984640000, 20708778572935434707683938140160000000, 302101709923756073800654275737927385319576932502732800000000000
Offset: 3

Views

Author

Amarnath Murthy, Apr 22 2004

Keywords

Examples

			a(4) = (1+2+3)*(1+2+4)*(1+3+4)*(2+3+4) = 3024.

References

  • Amarnath Murthy, Another combinatorial approach towards generalizing the AM-GM inequality, Octogon Mathematical Magazine Vol. 8, No. 2, October 2000.
  • Amarnath Murthy, Smarandache Dual Symmetric Functions And Corresponding Numbers Of The Type Of Stirling Numbers Of The First Kind. Smarandache Notions Journal Vol. 11, No. 1-2-3 Spring 2000.

Links

Crossrefs

Cf. A093883, A306594.

Programs

  • Mathematica
    Table[Product[(j + k + m), {k, 2, n}, {j, 1, k - 1}, {m, 1, j - 1}], {n, 3, 10}] (* Vaclav Kotesovec, Aug 31 2023 *)
Table[Product[Sqrt[BarnesG[3*k] * BarnesG[k+2] * Gamma[k/2 + 1] / Gamma[3*k/2]] / (BarnesG[2*k + 1] * 2^((k-1)/2)), {k, 1, n}], {n, 3, 10}] (* Vaclav Kotesovec, Aug 31 2023 *)

Formula

a(n) ~ sqrt(Pi/A) * 2^(5/12 - n/4 - n^2 - 2*n^3/3) * 3^(-1/6 - 7*n/24 + 3*n^3/4) * exp(1/24 - n/3 + 3*n^2/4 - 11*n^3/36 + zeta(3)/(48*Pi^2)) * n^(11/24 + n/3 - n^2/2 + n^3/6), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 31 2023

Extensions

More terms from Vladeta Jovovic, May 27 2004

A368722 a(n) = Product_{i=1..n, j=1..n, k=1..n} (i^3 + j^3 + k^3).

Original entry on oeis.org

1, 3, 353736000, 4795587079853800623303366670123008000000
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 04 2024

Keywords

Comments

Next term is too long to be included.

Crossrefs

Cf. A306594, A324425, A368723.
Cf. A324426, A368720.

Programs

  • Mathematica
    Table[Product[i^3 + j^3 + k^3, {i, 1, n}, {j, 1, n}, {k, 1, n}], {n, 0, 5}]

Formula

Limit_{n->oo} a(n)^(1/(n^3)) / n^3 = exp(Integral_{x=0..1, y=0..1, z=0..1} log(x^3 + y^3 + z^3) dz dy dx) = 0.5334736919092639993380174031245...

A368723 a(n) = Product_{i=1..n, j=1..n, k=1..n} (i^4 + j^4 + k^4).

Original entry on oeis.org

1, 3, 30180180096, 130911253854794147456410254996552949923277899497472
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 04 2024

Keywords

Comments

Next term is too long to be included.
In general, for m>0, limit_{n->oo} (Product_{i=1..n, j=1..n, k=1..n} (i^m + j^m + k^m))^(1/(n^3)) / n^m = exp(Integral_{x=0..1, y=0..1, z=0..1} log(x^m + y^m + z^m) dz dy dx) = exp(Integral_{x=0..1, y=0..1} (log(1 + x^k + y^k) - k + k*hypergeom2F1(1/k, 1, (k+1)/k, -1/(x^k + y^k))) dy dx).

Crossrefs

Cf. A306594 (m=1), A324425 (m=2), A368722 (m=3).
Cf. A324437, A368721.

Programs

  • Mathematica
    Table[Product[i^4 + j^4 + k^4, {i, 1, n}, {j, 1, n}, {k, 1, n}], {n, 0, 5}]

Formula

Limit_{n->oo} a(n)^(1/(n^3)) / n^4 = exp(Integral_{x=0..1, y=0..1, z=0..1} log(x^4 + y^4 + z^4) dz dy dx) = 0.3570458697635761757481417...

A325052 a(n) = Product_{i=0..n, j=0..n, k=0..n} (i! + j! + k!).

Original entry on oeis.org

3, 6561, 10319560704000000, 47749397192482757629144508002855841842593792000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 26 2019

Keywords

Comments

Next term is too long to be included.

Crossrefs

Cf. A306594, A306729, A324425, A325053.

Programs

  • Mathematica
    Table[Product[i! + j! + k!, {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 5}]
Clear[a]; a[n_] := a[n] = If[n == 0, 3, a[n-1] * Product[k! + j! + n!, {j, 0, n}, {k, 0, n}]^3 * (3*n!) / (Product[k! + 2*n!, {k, 0, n}]^3)]; Table[a[n], {n, 0, 5}]

Formula

a(n) ~ c * 2^(n^3/2 + 3*n^2 + 3*n) * 3^n * Pi^(n^3/2 + 3*n^2/2 + 3*n/2) * n^(3*n^4/4 + 3*n^3 + 17*n^2/4 + 5*n/2 + 601/120) / exp(15*n^4/16 + 3*n^3 + 3*n^2 - 21*n/4), where c = 28023.0953536911860317693532637428153075420958129597133...

A368685 a(n) = Product_{j=1..n, k=1..n} (j + k + n).

Original entry on oeis.org

1, 3, 600, 35562240, 1434015830016000, 70448433354492434841600000, 6610702315560389323908439364075520000000, 1709479709147705756603303596364188306401499545600000000000, 1660017838341811463102474357555838707949172571314554168163386261504000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 03 2024

Keywords

Crossrefs

Cf. A079478, A306594, A324444, A368622, A368686.

Programs

  • Mathematica
    Table[Product[i+j+n, {i, 1, n}, {j, 1, n}], {n, 0, 8}]
Join[{1}, Table[3*BarnesG[n] * BarnesG[3*n] * Gamma[n]^2 * Gamma[3*n]^2 / (4*BarnesG[2*n]^2 * Gamma[2*n]^4), {n, 1, 8}]]

Formula

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

A324427 a(n) = Product_{k=1..n} (Product_{j=1..k} (Product_{i=1..j} (i+j+k))).

Original entry on oeis.org

1, 3, 360, 38102400, 109506663383040000, 337878174593229551661219840000000, 54048023654871725380569225530796717972337459200000000000, 25571582464158460440549345359703385621119611033206432205259362823202406400000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 27 2019

Keywords

Crossrefs

Cf. A079478, A112332, A306594.

Programs

  • Maple
    a:= n-> mul(mul(mul(i+j+k, i=1..j), j=1..k), k=1..n):
seq(a(n), n=0..8);  # Alois P. Heinz, Jun 24 2023
  • Mathematica
    Table[Product[Product[Product[i+j+k, {i, 1, j}], {j, 1, k}], {k, 1, n}], {n, 0, 10}]
Table[Sqrt[Product[2^k Gamma[1 + 3*k/2]/Gamma[1 + k/2] (BarnesG[2 + k] BarnesG[2 + 3 k] )/BarnesG[2 + 2 k]^2 , {k, 1, n}]], {n, 0, 10}]
  • PARI
    a(n) = prod(k=1, n, prod(j=1, k, prod(i=1, j, i+j+k))); \\ Michel Marcus, Feb 27 2019

Formula

a(n) ~ 3^(3*n^3/4 + 9*n^2/4 + 47*n/24 + 7/24) * n^(n^3/6 + n^2/2 + n/3) / (2^(2*n^3/3 + 2*n^2 + 7*n/4 + 7/24) * exp(11*n^3/36 + 3*n^2/4 + n/3 - zeta(3)/(48*Pi^2))). - Vaclav Kotesovec, Nov 27 2023

Extensions

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

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

Original entry on oeis.org

1, 4, 2240421120000, 2357018782335863659143506877669927151046989269393693317529600000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 28 2019

Keywords

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

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

Views

Author

Vaclav Kotesovec, Jan 03 2024

Keywords

Crossrefs

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

Programs

  • Mathematica
    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}]]

Formula

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.
Showing 1-10 of 11 results. Next

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