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 1-7 of 7 results.

A306398 Decimal expansion of a constant related to the asymptotics of A324443.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Mar 01 2019

Keywords

Examples

			0.174039491910767235447561905910234481891384493843452148086901923847712649...

Links

Crossrefs

Cf. A324443.

Formula

Equals limit_{n->infinity} A324443(n) / (2^(n*(n+1)) * exp(Pi*n*(n+1)/2 - 3*n^2) * n^(2*n^2 + (Pi - 1)/2)).

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

A203511 a(n) = Product_{1 <= i < j <= n} (t(i) + t(j)); t = A000217 = triangular numbers.

Original entry on oeis.org

1, 1, 4, 252, 576576, 87178291200, 1386980110791475200, 3394352757964564324299571200, 1760578659300452732262852600316664217600, 255323290537547288382098619855584488593426606981120000
Offset: 0

Views

Author

Clark Kimberling, Jan 03 2012

Keywords

Comments

Each term divides its successor, as in A203512.
See A093883 for a guide to related sequences.

Crossrefs

Cf. A000217, A203512, A293290, A324403, A324443.

Programs

  • Maple
    t:= n-> n*(n+1)/2:
a:= n-> mul(mul(t(i)+t(j), i=1..j-1), j=2..n):
seq(a(n), n=0..12);  # Alois P. Heinz, Jul 23 2017
  • Mathematica
    f[j_] := j (j + 1)/2; z = 15;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]               (* A203511 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]  (* A203512 *)
Table[Product[k*(k+1)/2 + j*(j+1)/2, {k, 1, n}, {j, 1, k-1}], {n, 0, 10}] (* Vaclav Kotesovec, Sep 07 2023 *)

Formula

a(n) ~ c * 2^n * exp(n^2*(Pi/4 - 3/2) + n*(Pi/2 + 1)) * n^(n^2 - n - 2 - Pi/8), where c = 0.2807609661547466473998991675307759198889389396430915721129636653... - Vaclav Kotesovec, Sep 07 2023

Extensions

Name edited by Alois P. Heinz, Jul 23 2017
a(0)=1 prepended by Alois P. Heinz, Jul 29 2017

A307209 Decimal expansion of Product_{i>=1, j>=1} (1 + 1/(i^3 + j^3)).

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Mar 28 2019

Keywords

Comments

Product_{i>=1, j>=1} (1 + 1/(i^2 + j^2)) is divergent.
A324443(n) / A324403(n) ~ c * n^(Pi/2), where c = A306398 * 2^(3/4) * exp(-Pi/12) * Pi^(1/4) * Gamma(3/4) = 0.36753062884677326134620846786416595535234038999313...
Product_{i=1..n, j=1..n} (1 + 1/(i + j)) = A324444(n) / A079478(n) ~ 2^(2*n + 1) / (sqrt(Pi)*n^(3/2)).

Examples

			3.50478299933972837589112057043806125583893247862712753541994626614058385...

Links

Crossrefs

Cf. A073017, A307210, A307215, A324403, A324426, A324443.

Programs

  • Mathematica
    (* The iteration cycle: *) $MaxExtraPrecision = 1000; funs[n_] := Product[1 + 1/(i^3 + j^3), {i, 1, n}, {j, 1, n}]; Do[Print[N[Sum[(-1)^(m + j)*funs[j*Floor[200/m]] * j^(m - 1)/(j - 1)!/(m - j)!, {j, 1, m}], 100]], {m, 10, 100, 10}]
  • PARI
    default(realprecision, 50); exp(sumalt(k=1, -(-1)^k/k*sumnum(i=1, sumnum(j=1, 1/(i^3+j^3)^k)))) \\ 15 decimals correct

Formula

Equals limit_{n->infinity} A307210(n) / A324426(n).

A307210 a(n) = Product_{i=1..n, j=1..n} (i^3 + j^3 + 1).

Original entry on oeis.org

1, 3, 5100, 305727048000, 7748770873210669158912000, 476007332700693200670745550306381336371200000, 272661655519533773844144991586798737775635133552905539740860416000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 28 2019

Keywords

Comments

Product_{i=1..n, j=1..n} (1 + 1/(i + j)) = A324444(n) / A079478(n) ~ 2^(2*n + 1) / (sqrt(Pi)*n^(3/2)).
Product_{i=1..n, j=1..n} (1 + 1/(i^2 + j^2)) = A324443(n) / A324403(n) ~ c * n^(Pi/2), where c = A306398 * 2^(3/4) * exp(-Pi/12) * Pi^(1/4) * Gamma(3/4) = 0.36753062884677326134620846786416595535234038999313315993144237973600...

Crossrefs

Cf. A307209, A324403, A324426, A324443, A324444.

Programs

  • Maple
    a:= n-> mul(mul(i^3+j^3+1, i=1..n), j=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023
  • Mathematica
    Table[Product[i^3 + j^3 + 1, {i, 1, n}, {j, 1, n}], {n, 1, 8}]

Formula

a(n) ~ A307209 * A324426(n).
a(n) ~ c * A * 2^(2*n*(n+1) + 1/4) * exp(Pi*(n*(n+1) + 1/6)/sqrt(3) - 9*n^2/2 - 1/12) * n^(3*n^2 - 3/4) / (3^(5/6) * Pi^(1/6) * Gamma(2/3)^2), where c = A307209 = Product_{i>=1, j>=1} (1 + 1/(i^3 + j^3)) = 3.504782999339728375891120570... and A is the Glaisher-Kinkelin constant A074962.

Extensions

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

A307215 Decimal expansion of Product_{i>=1, j>=1} (1 + 1/(i^4 + j^4)).

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Mar 29 2019

Keywords

Comments

Product_{i=1..n, j=1..n} (1 + 1/(i + j)) = A324444(n) / A079478(n) ~ 2^(2*n + 1) / (sqrt(Pi)*n^(3/2)).
Product_{i=1..n, j=1..n} (1 + 1/(i^2 + j^2)) = A324443(n) / A324403(n) ~ c * n^(Pi/2), where c = A306398 * 2^(3/4) * exp(-Pi/12) * Pi^(1/4) * Gamma(3/4) = 0.36753062884677326134620846786416595535234038999313315993144237973600...
Product_{i>=1, j>=1} (1 + 1/(i^3 + j^3)) = A307209 = 3.50478299933972837589112...

Examples

			1.94073028537236152995386077599647772038707968293217092813061397472522642172...

Crossrefs

Cf. A307209, A324437.

Programs

  • Mathematica
    (* The iteration cycle: *) $MaxExtraPrecision = 1000; funs[n_] := Product[1 + 1/(i^4 + j^4), {i, 1, n}, {j, 1, n}]; Do[Print[N[Sum[(-1)^(m + j)*funs[j*Floor[200/m]] * j^(m - 1)/(j - 1)!/(m - j)!, {j, 1, m}], 100]], {m, 10, 100, 10}]

Formula

Equals limit_{n->infinity} (Product_{i=1..n, j=1..n} (1 + i^4 + j^4)) / A324437(n).

A306728 a(n) = Product_{i=1..n, j=1..n} (i*(i+1) + j*(j+1)).

Original entry on oeis.org

1, 4, 3072, 4682022912, 62745927042654535680, 22033340103629170301586112512000000, 479715049773154880180722813201712394999926095872000000, 1318058833735625830065875826842622254472987373414662267314001234660163584000000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 06 2019

Keywords

Crossrefs

Cf. A324403, A324443.

Programs

  • Mathematica
    Table[Product[i*(i+1)+j*(j+1), {i, 1, n}, {j, 1, n}], {n, 0, 8}]
  • Python
    from math import prod, factorial
def A306728(n): return (prod(i*(i+1)+j*(j+1) for i in range(1,n) for j in range(i+1,n+1))*factorial(n))**2*(n+1)<Chai Wah Wu, Nov 22 2023

Formula

a(n) ~ c * 2^(n*(n+2)) * exp(Pi*n*(n+2)/2 - 3*n^2) * n^(2*n^2 - 2 - Pi/4), where c = 0.4952828896469310726828820344381813905230827930914109676983577406850360879...
Showing 1-7 of 7 results.

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