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 1-4 of 4 results.

A367567 a(n) = Product_{k=0..n} (3*k)! / k!^3.

Original entry on oeis.org

1, 6, 540, 907200, 31434480000, 23788231346880000, 408042767492495815680000, 162838835029822082951032012800000, 1541352909587869227178909850805190656000000, 351233376660297011570511252132131832794456064000000000, 1949695346852822356399298814748829537555898997004605685760000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 23 2023

Keywords

Crossrefs

Cf. A000178, A006480, A268504, A061719, A268196.
Cf. A007685, A367568, A367569, A367570, A367571.

Programs

  • Mathematica
    Table[Product[(3*k)!/k!^3, {k, 0, n}], {n, 0, 10}]
Table[Product[Binomial[3*k,k] * Binomial[2*k,k], {k, 0, n}], {n, 0, 10}]

Formula

a(n) = Product_{k=0..n} binomial(3*k,k) * binomial(2*k,k).
a(n) = A268504(n) / A000178(n)^3.
a(n) = A268504(n) / A061719(n).
a(n) = A007685(n) * A268196(n).
a(n) ~ A^(8/3) * Gamma(1/3)^(1/3) * 3^(3*n^2/2 + 2*n + 11/36) * exp(n - 2/9) / (n^(n + 13/18) * (2*Pi)^(n + 7/6)), where A is the Glaisher-Kinkelin constant A074962.

A367569 a(n) = Product_{k=0..n} (5*k)! / k!^5.

Original entry on oeis.org

1, 120, 13608000, 2288430144000000, 699207483978843840000000000, 435858496811697532778806061260800000000000, 597507154003470929939550139366865942134606725120000000000000, 1898554530971015145216561379837863419725314413457243266261094236160000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 23 2023

Keywords

Crossrefs

Cf. A000178, A008978, A268506.
Cf. A007685, A367567, A367568, A367570, A367571.

Programs

  • Mathematica
    Table[Product[(5*k)!/k!^5, {k, 0, n}], {n, 0, 10}]
Table[Product[Binomial[5*k,k] * Binomial[4*k,k] * Binomial[3*k,k] * Binomial[2*k,k], {k, 0, n}], {n, 0, 10}]

Formula

a(n) = Product_{k=0..n} binomial(5*k,k) * binomial(4*k,k) * binomial(3*k,k) * binomial(2*k,k).
a(n) = A268506(n) / A000178(n)^5.
a(n) ~ A^(24/5) * Gamma(1/5)^(3/5) * Gamma(2/5)^(2/5) * Gamma(3/5)^(1/5) * 5^(5*n^2/2 + 3*n + 23/60) * exp(2*n - 2/5) / (n^(2*n + 7/5) * (2*Pi)^(2*n + 13/5)), where A is the Glaisher-Kinkelin constant A074962.
Equivalently, a(n) ~ A^(24/5) * Gamma(1/5)^(3/5) * Gamma(2/5)^(1/5) * 5^(5*n^2/2 + 3*n + 1/3) * exp(2*n - 2/5) / ((1 + sqrt(5))^(1/10) * 2^(2*n + 23/10) * Pi^(2*n + 12/5) * n^(2*n + 7/5)).

A367570 a(n) = Product_{k=0..n} (6*k)! / k!^6.

Original entry on oeis.org

1, 720, 5388768000, 739474163011584000000, 2400828978003787120431882240000000000, 213271990853093812884314351984207293234859212800000000000, 569474121824212834327144127568532894901251393782268174537457286512640000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 23 2023

Keywords

Crossrefs

Cf. A000178, A008979, A271946.
Cf. A007685, A367567, A367568, A367569, A367571.

Programs

  • Mathematica
    Table[Product[(6*k)!/k!^6, {k, 0, n}], {n, 0, 10}]
Table[Product[Binomial[6*k,k] * Binomial[5*k,k] * Binomial[4*k,k] * Binomial[3*k,k] * Binomial[2*k,k], {k, 0, n}], {n, 0, 10}]

Formula

a(n) = Product_{k=0..n} binomial(6*k,k) * binomial(5*k,k) * binomial(4*k,k) * binomial(3*k,k) * binomial(2*k,k).
a(n) = A271946(n) / A000178(n)^6.
a(n) ~ A^(35/6) * Gamma(1/3)^(5/3) * 2^(3*n^2 + n - 215/72) * 3^(3*n^2 + 7*n/2 + 47/72) * exp(5*n/2 - 35/72) / (n^(5*n/2 + 125/72) * Pi^(5*n/2 + 10/3)), where A is the Glaisher-Kinkelin constant A074962.

A367571 a(n) = Product_{k=0..n} (7*k)! / k!^7.

Original entry on oeis.org

1, 5040, 3432645216000, 626489905645044080640000000, 41646279370357699257014919153469440000000000000, 1200992054275801322636044235924808416678612164215512865177600000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 23 2023

Keywords

Comments

In general, for m > 1, Product_{k=0..n} (m*k)! / k!^m ~ A^(m - 1/m) * exp(m*n/2 - m/12 + 1/(12*m) - n/2) * m^(m*n^2/2 + m*n/2 - 1/(12*m) + n/2) * n^(-m*n/2 - m/3 + 1/(12*m) + n/2 + 1/4) * (2*Pi)^(-m*n/2 - m/4 + n/2 + 1/4) / Product_{j=1..m-1} Gamma(j/m)^(j/m), where A is the Glaisher-Kinkelin constant A074962.

Crossrefs

Cf. A000178, A071549, A271947.
Cf. A007685, A367567, A367568, A367569, A367570.

Programs

  • Mathematica
    Table[Product[(7*k)!/k!^7, {k, 0, n}], {n, 0, 10}]
Table[Product[Binomial[7*k,k] * Binomial[6*k,k] * Binomial[5*k,k] * Binomial[4*k,k] * Binomial[3*k,k] * Binomial[2*k,k], {k, 0, n}], {n, 0, 10}]

Formula

a(n) = Product_{k=0..n} binomial(7*k,k) * binomial(6*k,k) * binomial(5*k,k) * binomial(4*k,k) * binomial(3*k,k) * binomial(2*k,k).
a(n) = A271947(n) / A000178(n)^7.
a(n) ~ A^(48/7) * 7^(7*n^2/2 + 4*n - 1/84) * exp(3*n - 4/7) / (Gamma(1/7)^(1/7) * Gamma(2/7)^(2/7) * Gamma(3/7)^(3/7) * Gamma(4/7)^(4/7) * Gamma(5/7)^(5/7) * Gamma(6/7)^(6/7) * n^(3*n + 29/14) * (2*Pi)^(3*n + 3/2)), where A is the Glaisher-Kinkelin constant A074962.
Showing 1-4 of 4 results.

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