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.

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A203589 Vandermonde sequence using x^2 + y^2 applied to (1,3,5,...,2n-1).

Original entry on oeis.org

1, 10, 8840, 1897064000, 192924579369600000, 15340654595434137315840000000, 1423341281300698059502838358528000000000000, 215088732628531399592688671811428988579913728000000000000000
Offset: 1

Views

Author

Clark Kimberling, Jan 04 2012

Keywords

Comments

See A093883 for a discussion and guide to related sequences.

Crossrefs

Cf. A293290, A324403.

Programs

  • Mathematica
    f[j_] := 2 j - 1; z = 12;
v[n_] := Product[Product[f[j]^2 + f[k]^2, {j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203589 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203590 *)
Table[Product[(2*k - 1)^2 + (2*j - 1)^2, {k, 1, n}, {j, 1, k-1}], {n, 1, 10}] (* Vaclav Kotesovec, Sep 08 2023 *)

Formula

a(n) ~ 2^(3*n^2/2 - 3*n/2 - 3/8) * n^(n*(n-1)) / exp((6 - Pi)*n^2/4 - n + Pi/48). - Vaclav Kotesovec, Sep 08 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).

A367941 a(n) = Product_{i=1..n, j=1..n} (i^2 + 2*j^2).

Original entry on oeis.org

1, 3, 1944, 4102777008, 140890630179993255936, 247470977313135626800897828778803200, 54132901224855040835735917614114353691165557521593139200
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 05 2023

Keywords

Crossrefs

Cf. A324403, A367942, A367943, A367944.
Cf. A367542, A367543.

Programs

  • Mathematica
    Table[Product[i^2+2*j^2, {i, 1, n}, {j, 1, n}], {n, 0, 8}]

Formula

a(n) ~ c * n^(2*n^2 - 1/2) * 3^(n*(n+1)) * 2^(-n/2) * exp(n*(n+1)*(Pi - arctan(sqrt(2))) / sqrt(2) - 3*n^2) , where c = 0.4690673220228472212446336926899602910226601891141458824921925169726804439...

A367942 a(n) = Product_{i=1..n, j=1..n} (i^2 + 3*j^2).

Original entry on oeis.org

1, 4, 5824, 45861064704, 9751658280030585225216, 176005320076923781520069562958715289600, 656508955366282248103393001602851493819854909361664242483200
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 05 2023

Keywords

Crossrefs

Cf. A324403, A367941, A367943, A367944.
Cf. A367542, A367543.

Programs

  • Mathematica
    Table[Product[i^2+3*j^2, {i, 1, n}, {j, 1, n}], {n, 0, 8}]

Formula

a(n) ~ c * n^(2*n^2 - 1/2) * 4^(n*(n+1)) * 3^(-n/2) * exp(5*Pi*n*(n+1)/(6*sqrt(3)) - 3*n^2), where c = 0.4612030005343304845802441101292774353695846313857765074861837886133930626...

A367943 a(n) = Product_{i=1..n, j=1..n} (i^2 + 4*j^2).

Original entry on oeis.org

1, 5, 13600, 294372000000, 252880261890048000000000, 27099784799070466617992871936000000000000, 882065676199020188908312950703217787436793856000000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 05 2023

Keywords

Crossrefs

Cf. A324403, A367941, A367942, A367944.
Cf. A367542, A367543.

Programs

  • Mathematica
    Table[Product[i^2+4*j^2, {i, 1, n}, {j, 1, n}], {n, 0, 8}]

Formula

a(n) ~ c * n^(2*n^2 - 1/2) * 5^(n*(n+1)) * 2^(-n) * exp(n*(n+1)*(2*Pi - 3*arctan(2))/2 - 3*n^2) , where c = 0.4523180383519335764034720087114905921141637339852374451758854101884791581...

A368064 a(n) = Product_{i=1..n, j=1..n} (i^2 + 4*i*j + j^2).

Original entry on oeis.org

1, 6, 24336, 870746557824, 1311726482483997806493696, 256433546267136937832915286844640487014400, 15678550451426175377500759401206644047210595564950427820202393600
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 10 2023

Keywords

Crossrefs

Cf. A367543, A324403, A367542, A079478^2, A368067, A368065, A368066.

Programs

  • Mathematica
    Table[Product[i^2 + 4*i*j + j^2, {i, 1, n}, {j, 1, n}], {n, 0, 7}]

Formula

a(n) ~ 2^((3+sqrt(3))*n*(n+1) + (sqrt(3)-1)/6) * 3^(3*n*(n+1) + 13/24) * n^(2*n^2 - 7/6) / (Gamma(1/3)^(1/2) * Gamma(1/4)^(1/3) * Pi^(7/12) * (1 + sqrt(3))^((6*n*(n+1) + 1)/sqrt(3) - 1/2) * exp(3*n^2)).

A368067 a(n) = Product_{i=1..n, j=1..n} (i^2 + 3*i*j + j^2).

Original entry on oeis.org

1, 5, 12100, 188898484500, 91554454518735288960000, 4263420404009649597344435073399120000000, 46073465749493255153019723901007197815549903333795840000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 10 2023

Keywords

Crossrefs

Cf. A367543, A324403, A367542, A079478^2, A368064, A368065, A368066.

Programs

  • Mathematica
    Table[Product[i^2 + 3*i*j + j^2, {i, 1, n}, {j, 1, n}], {n, 0, 7}]

Formula

a(n) ~ 5^(5*n*(n+1)/2 + 1/2) * n^(2*n^2 - 1) / (2 * Pi * exp(3*n^2) * phi^(sqrt(5)*(n*(n+1) + 1/6) - 1/2)), where phi = A001622 is the golden ratio.

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...
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