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 1-9 of 9 results.

A306617 Decimal expansion of a constant related to the asymptotics of A324425.

Original entry on oeis.org

8, 2, 8, 8, 5, 9, 5, 7, 9, 6, 6, 9, 2, 7, 9, 2, 8, 6, 6, 9, 7, 2, 2, 9, 0, 2, 0, 7, 7, 5, 1, 0, 3, 0, 2, 6, 7, 6, 9, 1, 0, 5, 7, 5, 5, 9, 7, 7, 1, 2, 1, 1, 4, 5, 2, 4, 4, 0, 4, 0, 3, 3, 1, 7, 9, 5, 7, 1, 8, 3, 4, 3, 0, 2, 2, 1, 4, 7, 1, 8, 3, 7, 7, 6, 7, 1, 1, 3, 1, 1, 8, 9, 2, 7, 8, 7, 3, 0, 4, 0, 5, 4, 9, 3, 0, 9
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 28 2019

Keywords

Comments

Ulrich Neumann found a closed form, see the "Mathematica Stack Exchange" link.

Examples

			0.828859579669279286697229020775103026769105755977121145244040331795...

Links

Crossrefs

Cf. A324425.

Programs

  • Maple
    evalf(exp(integrate(log(x^2 + y^2 + z^2), x = 0..1, y = 0..1, z = 0..1)), 20);
evalf(exp(integrate(-2 + 2*sqrt(y^2 + z^2) * arctan(1/sqrt(y^2 + z^2)) + log(1 + y^2 + z^2), y = 0..1, z = 0..1)), 20);
  • Mathematica
    ixr = Exp[Integrate[1/3 (Log[1 + Sec[fi]^2] + (-7 + 3 Log[1 + Sec[fi]^2]) Sec[fi]^2 + 2 (Pi - 2 ArcTan[Sec[fi]]) Sec[fi]^3), {fi, 0, Pi/4}]]; Chop[N[ixr, 120]] (* A program by Ulrich Neumann added by Vaclav Kotesovec, Mar 03 2019. The calculation takes several minutes. *)
  • PARI
    exp(intnum(z=0, 1 ,intnum(y=0, 1, intnum(x=0, 1, log(x^2 + y^2 + z^2)))))
  • PARI
    exp(intnum(z=0, 1 ,intnum(y=0, 1, -2 + 2*sqrt(y^2 + z^2) * atan(1/sqrt(y^2 + z^2)) + log(1 + y^2 + z^2))))

Formula

Equals limit_{n->infinity} (A324425(n)^(1/n^3))/n^2.

Extensions

More terms computed by Ulrich Neumann added by Vaclav Kotesovec, Mar 03 2019

A324403 a(n) = Product_{i=1..n, j=1..n} (i^2 + j^2).

Original entry on oeis.org

1, 2, 400, 121680000, 281324160000000000, 15539794609114833408000000000000, 49933566483104048708063697937367040000000000000000, 19323883089768863178599626514889213871887405416448000000000000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 26 2019

Keywords

Comments

Next term is too long to be included.

Crossrefs

Cf. A079478, A324426, A324437, A324438, A324439, A324440, A367834.
Cf. A272244, A293290, A324402, A324443, A324425.

Programs

  • Maple
    a:= n-> mul(mul(i^2+j^2, i=1..n), j=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023
  • Mathematica
    Table[Product[i^2+j^2, {i, 1, n}, {j, 1, n}], {n, 1, 10}]
  • PARI
    a(n) = prod(i=1, n, prod(j=1, n, i^2+j^2)); \\ Michel Marcus, Feb 27 2019
  • Python
    from math import prod, factorial
def A324403(n): return (prod(i**2+j**2 for i in range(1,n) for j in range(i+1,n+1))*factorial(n))**2<Chai Wah Wu, Nov 22 2023

Formula

a(n) ~ 2^(n*(n+1) - 3/4) * exp(Pi*n*(n+1)/2 - 3*n^2 + Pi/12) * n^(2*n^2 - 1/2) / (Pi^(1/4) * Gamma(3/4)).
a(n) = 2*n^2*a(n-1)*Product_{i=1..n-1} (n^2 + i^2)^2. - Chai Wah Wu, Feb 26 2019
For n>0, a(n)/a(n-1) = A272244(n)^2 / (2*n^6). - Vaclav Kotesovec, Dec 02 2023
a(n) = exp(2*Integral_{x=0..oo} (n^2/(x*exp(x)) - (cosh(n*x) - cos(n*x))/(x*exp((n + 1)*x)*(cosh(x) - cos(x)))) dx)/2^(n^2). - Velin Yanev, Jun 30 2025

Extensions

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

A306594 a(n) = Product_{i=1..n, j=1..n, k=1..n} (i + j + k).

Original entry on oeis.org

1, 3, 144000, 455282248974336000000, 9608917807566747651759509633033255126040576000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 27 2019

Keywords

Comments

Next term is too long to be included.

Crossrefs

Cf. A079478, A093884, A112332, A324425, A368722, A368723, A324441.

Programs

  • Maple
    a:= n-> mul(mul(mul(i+j+k, 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+j+k, {i, 1, n}, {j, 1, n}, {k, 1, n}], {n, 1, 6}]
Table[Product[k^(3*(n - k + 1) (n - k + 2)/2), {k, 1, n}] * Product[k^((3*n - k + 1) (3*n - k + 2)/2), {k, 1, 3*n}] / Product[k^(3*(2*n - k + 1) (2*n - k + 2)/2), {k, 1, 2*n}], {n, 1, 6}]
Clear[a]; a[n_] := a[n] = If[n == 1, 3, 3*n*a[n-1] * BarnesG[2+n]^3 * BarnesG[2+3*n]^3 * Gamma[1+2*n]^3 / (BarnesG[2+2*n]^6 * Gamma[1+3*n]^3)]; Table[a[n], {n, 1, 6}] (* Vaclav Kotesovec, Mar 28 2019 *)

Formula

a(n) = Product_{k=1..n} (BarnesG(k+2) * BarnesG(2*n+k+2) / BarnesG(n+k+2)^2).
a(n) = Product_{k=1..n} (k^(3*(n - k + 1)*(n - k + 2)/2)) * Product_{k=1..3*n} (k^((3*n - k + 1)*(3*n - k + 2)/2)) / Product_{k=1..2*n} (k^(3*(2*n - k + 1)*(2*n - k + 2)/2)).
a(n) ~ sqrt(Pi) * 3^(9*n^3/2 + 27*n^2/4 + 3*n + 3/8) * n^(n^3 + 3/8) / (A^(3/2) * 2^(4*n^3 + 9*n^2 + 6*n + 5/8) * exp(11*n^3/6 - Zeta(3)/(8*Pi^2) - 1/8)), where A is the Glaisher-Kinkelin constant A074962.

Extensions

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

A324443 a(n) = Product_{i=1..n, j=1..n} (1 + i^2 + j^2).

Original entry on oeis.org

1, 3, 972, 437987088, 1396064690700615936, 100943980553724942717460016640000, 408685260379151918936869901376463191556211834880000, 193581283410907012468703321819613695893448022144552623141894180044800000000
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 28 2019

Keywords

Comments

Product_{i>=1, j>=1} (1 + 1/(i^2 + j^2)) is divergent.

Crossrefs

Cf. A101686, A324403, A324425, A324444, A306398.

Programs

  • Maple
    a:= n-> mul(mul(1+i^2+j^2, i=1..n), j=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023
  • Mathematica
    Table[Product[1 + i^2 + j^2, {i, 1, n}, {j, 1, n}], {n, 1, 10}]

Formula

a(n) ~ c * 2^(n*(n+1)) * exp(Pi*n*(n+1)/2 - 3*n^2) * n^(2*n^2 + (Pi - 1)/2), where c = A306398 = 0.1740394919107672354475619059102344818913844938434521480869...
a(n) / A324403(n) ~ d * n^(Pi/2), where d = A306398 * 2^(3/4) * exp(-Pi/12) * Pi^(1/4) * Gamma(3/4) = 0.36753062884677326134620846786416595535234038999313...

Extensions

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

A368622 a(n) = Product_{j=1..n, k=1..n} (j^2 + k^2 + n^2).

Original entry on oeis.org

1, 3, 5832, 172907569296, 419358815743567702818816, 267800543010963952830647446563000000000000, 110831581527076064529150462985910455129725821244148698662830080000
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 01 2024

Keywords

Comments

The limit has a closed form. In Mathematica: Exp[Integrate[Log[x^2 + y^2 + 1], {x,0,1}, {y,0,1}]]. The output is extremely large.

Crossrefs

Cf. A272244, A324403, A324425, A368720, A368721.

Programs

  • Mathematica
    Table[Product[j^2 + k^2 + n^2, {j, 1, n}, {k, 1, n}], {n, 0, 10}]

Formula

Limit_{n->oo} a(n)^(1/(n^2)) / n^2 = exp(Integral_{x=0..1, y=0..1} log(x^2 + y^2 + 1) dy dx) = 1.6143980185761253961882683158432481977126507900460725431661...

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...

A368623 a(n) = Product_{k=1..n} (k^2 + 2*n^2).

Original entry on oeis.org

1, 3, 108, 11286, 2337984, 804305700, 414285404544, 298436020283016, 286455044544970752, 353358684943164351792, 544692796454778554880000, 1025983872949208210500475232, 2318663822077115453077590638592, 6191980828123077577798830642106944, 19289639610614384872295428226588737536
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 01 2024

Keywords

Comments

In general, for d>0, Product_{k=1..n} (k^2 + d*n^2) ~ (d+1)^(n + 1/2) * exp(n*(sqrt(d)*(Pi - 2*arctan(sqrt(d))) - 2)) * n^(2*n) / sqrt(d). - Vaclav Kotesovec, Jan 06 2024

Crossrefs

Cf. A272244, A324403, A324425.

Programs

  • Mathematica
    Table[Product[k^2 + 2*n^2, {k, 1, n}], {n, 0, 20}]

Formula

a(n) ~ 3^(n + 1/2) * exp(n*(sqrt(2)*arctan(2*sqrt(2)) - 2)) * n^(2*n) / sqrt(2).
Showing 1-9 of 9 results.

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