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.

A306760 a(n) = Product_{i=1..n, j=1..n} (i*j + 1).

Original entry on oeis.org

1, 2, 90, 705600, 4105057320000, 52487876090562232320000, 3487017405172854771910634342400000000, 2448893405298238642974553493547144534294528000000000000, 33257039167768610289435138215602132823918399655132218973388800000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 08 2019

Keywords

Crossrefs

Cf. A079478, A101686, A185141, A324444, A306765, A324589, A306907.

Programs

  • 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[i*j + 1, {i, 1, n}, {j, 1, n}], {n, 1, 10}]
Table[n!^(2*n) * Product[Binomial[n + 1/j, n], {j, 1, n}], {n, 1, 10}]

Formula

a(n) ~ c * n^(n*(2*n+1) + 2*gamma) * (2*Pi)^n * exp(1/6 + log(n)^2 - 2*n^2), where c = 1/A306765 and gamma is the Euler-Mascheroni constant A001620.

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

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).

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)).

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).

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 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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