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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A057528 5th level factorials: product of first n 4th level factorials.

Original entry on oeis.org

1, 1, 2, 96, 31850496, 2524286414780230533120, 1189172215782988266980141580906985588465965465600000
Offset: 0

Views

Author

Henry Bottomley, Sep 02 2000

Keywords

Comments

In general for k-th level factorials a(n) =Product of first n (k-1)-th level factorials =Product[i^C(n-i+k-1,n-i)] over 1<=i<=n.

Crossrefs

Cf. A000142, A000178, A055462, A057527, A260404 for first, second, third, fourth and sixth level factorials.

Programs

  • Mathematica
    Table[Product[i^Binomial[n-i+4,4],{i,1,n}],{n,0,10}] (* Vaclav Kotesovec, Jul 24 2015 *)
Nest[FoldList[Times,#]&,Range[0,10]!,4] (* Harvey P. Dale, Dec 15 2021 *)

Formula

a(n) =a(n-1)*A057527(n) =Product[i^A000292(n-i+4)] over 1<=i<=n.
a(n) ~ exp(25/144 - 109*n/144 - 35*n^2/24 - 379*n^3/432 - 125*n^4/576 - 137*n^5/7200 + (35 + 30*n + 6*n^2)*Zeta(3)/(96*Pi^2) - Zeta(5)/(32*Pi^4) + (5+2*n)*Zeta'(-3)/12) * n^((5+2*n)*(19/288 + 25*n/144 + 5*n^2/36 + n^3/24 + n^4/240)) * (2*Pi)^((n+1)*(n+2)*(n+3)*(n+4)/48) / A^((5+2*n)*(5 + 5*n + n^2)/12), where Zeta(3) = A002117, Zeta(5) = A013663, Zeta'(-3) = A259068 = 0.00537857635777430114441697421... and A = A074962 = 1.282427129100622636875... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 24 2015

A260404 6th level factorials: product of first n 5th level factorials.

Original entry on oeis.org

1, 1, 2, 192, 6115295232, 15436756676507918107049554083840, 18356962141505758798331790171539976807981714702571497465907439808868887035904000000
Offset: 0

Views

Author

Vaclav Kotesovec, Jul 24 2015

Keywords

Comments

In general for k-th level factorials a(n) = Product_{j=1..n} j^C(n-j+k-1,k-1).

Crossrefs

Cf. A000142, A000178, A055462, A057527, A057528.

Programs

  • Mathematica
    Table[Product[i^Binomial[n-i+5,5],{i,1,n}],{n,0,10}]

Formula

a(n) ~ exp(137/720 - 11*n/16 - 737*n^2/480 - 53*n^3/48 - 421*n^4/1152 - 137*n^5/2400 - 49*n^6/14400 + (3 + n)*(15 + 12*n + 2*n^2)*Zeta(3)/(96*Pi^2) - (3 + n)*Zeta(5) / (32*Pi^4) + (17 + 12*n + 2*n^2)*Zeta'(-3)/24 + Zeta'(-5)/120) * n^(19087/60480 + n + 137*n^2/120 + 5*n^3/8 + 17*n^4/96 + n^5/40 + n^6/720) * (2*Pi)^((n+1)*(n+2)*(n+3)*(n+4)*(n+5)/240) / A^(137/60 + 15*n/4 + 17*n^2/8 + n^3/2 + n^4/24), where Zeta(3) = A002117, Zeta(5) = A013663, Zeta'(-3) = A259068, Zeta'(-5) = A259070 and A = A074962 is the Glaisher-Kinkelin constant.

A306729 a(n) = Product_{i=0..n, j=0..n} (i! + j!).

Original entry on oeis.org

2, 16, 5184, 9559130112, 109045776752640000000000, 27488263744928988967331390258832998400000000000, 1147897050240877062218236820013018349788772091106840426434074807527014400000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 06 2019

Keywords

Crossrefs

Cf. A000178, A055462, A079478, A203482, A217757, A323717, A324403, A325052.

Programs

  • Mathematica
    Table[Product[i! + j!, {i, 0, n}, {j, 0, n}], {n, 0, 7}]
Clear[a]; a[n_] := a[n] = If[n == 0, 2, a[n-1] * Product[k! + n!, {k, 0, n}]^2 / (2*n!)]; Table[a[n], {n, 0, 7}] (* Vaclav Kotesovec, Mar 27 2019 *)
Table[Product[Product[k! + j!, {k, 0, j}], {j, 1, n}]^2 / (2^(n-1) * BarnesG[n + 2]), {n, 0, 7}] (* Vaclav Kotesovec, Mar 27 2019 *)
  • Python
    from math import prod, factorial as f
def a(n): return prod(f(i)+f(j) for i in range(n) for j in range(n))
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Feb 16 2021

Formula

a(n) ~ c * 2^(n^2/2 + 2*n) * Pi^(n^2/2 + n) * n^(2*n^3/3 + 2*n^2 + 11*n/6 + 5/2) / exp(8*n^3/9 + 2*n^2 + n), where c = A324569 = 62.14398692334529025548974541735...
a(n) = a(n-1) * A323717(n)^2 / (2*n!). - Vaclav Kotesovec, Mar 28 2019

A057527 4th level factorials: product of first n superduperfactorials.

Original entry on oeis.org

1, 1, 2, 48, 331776, 79254226206720, 471092427871945743012986880000, 351177419973413722592573060611594181593855426560000000000
Offset: 0

Views

Author

Henry Bottomley, Sep 02 2000

Keywords

Comments

In general for k-th level factorials a(n) =Product of first n (k-1)-th level factorials =Product[i^C(n-i+k-1,n-i)] over 1<=i<=n.

Examples

			a(4) =((4!*3!*2!*1!)*(3!*2!*1!)*(2!*1!)*(1!)) * ((3!*2!*1!)*(2!*1!)*(1!)) * ((2!*1!)*(1!)) * ((1!)) =24*6^3*2^6*1^10 =331776

Crossrefs

Cf. A000142, A000178, A055462, A057528, A260404 for first, second, third, fifth and sixth level factorials.

Programs

  • Mathematica
    Table[Product[i^Binomial[n-i+3,3],{i,1,n}],{n,0,10}] (* Vaclav Kotesovec, Jul 24 2015 *)
Nest[FoldList[Times,#]&,Range[0,8]!,3] (* Harvey P. Dale, Jan 08 2024 *)

Formula

a(n) =a(n-1)*A055462(n) =Product[i^A000332(n-i)] over 1<=i<=n.
a(n) ~ exp(11/72 - 5*n/6 - 4*n^2/3 - 11*n^3/18 - 25*n^4/288 + Zeta(3)*(n+2) / (8*Pi^2) + Zeta'(-3)/6) * n^(251/720 + n + 11*n^2/12 + n^3/3 + n^4/24) * (2*Pi)^((n+1)*(n+2)*(n+3)/12) / A^(11/6 + 2*n + n^2/2), where Zeta(3) = A002117, Zeta'(-3) = A259068 = 0.0053785763577743011444169742104138428956644397... and A = A074962 = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 24 2015

A306651 a(n) = Product_{k=1..n} BarnesG(3*k).

Original entry on oeis.org

1, 288, 36118462464000, 240498631970530185123135341199360000000000
Offset: 1

Views

Author

Vaclav Kotesovec, Mar 03 2019

Keywords

Comments

Next term is too long to be included.

Links

Crossrefs

Cf. A055462, A268504, A296608, A306635, A324993.

Programs

  • Mathematica
    Table[Product[BarnesG[3*k], {k, 1, n}], {n, 1, 6}]
Round[Table[3^(15*n^2/4 - 7*n/12 - 1/4) * E^(Pi/(18*Sqrt[3]) - PolyGamma[1, 1/3]/(12*Sqrt[3]*Pi) - Zeta[3]/(3*Pi^2) + 1/6 + 3*n*(n + 1)*(2*n + 1)/8 + 3*PolyGamma[-3, n + 1] - (3/2)*Derivative[1, 0][Zeta][-2, n] + (1/6)*Derivative[1, 0][Zeta][-2, 3*n] + (7/2)*Derivative[1, 0][Zeta][-1, n + 1/3] + (5/2)*Derivative[1, 0][Zeta][-1, n + 2/3]) * BarnesG[3*n]^(n + 1) * BarnesG[n + 1/3] * Gamma[n]^(5*n/2 - 13/6) / (BarnesG[4/3] * BarnesG[n]^(5/2) * Gamma[n + 1/3]^(n - 1) * Gamma[3*n]^(3*n*(n + 1)/2 - 2/3) * Glaisher^(3*n + 5) * (2*Pi)^(3*(n + 1)^2/4) * n^(3*n^2/2)), {n, 1, 6}]] (* Vaclav Kotesovec, Mar 04 2019 *)

Formula

a(n) ~ (2*Pi)^(3*n^2/4 + n/4 + 1/6) * 3^(3*n^3/2 + 3*n^2/4 - n/3 - 13/72) * n^(3*n^3/2 + 3*n^2/4 - n/3 - 5/72) / (Gamma(1/3)^(1/3) * A^(n + 1/6) * exp(11*n^3/4 + 9*n^2/8 - 5*n/12 - Zeta(3)/(24*Pi^2) - 1/72)), where A is the Glaisher-Kinkelin constant A074962.
a(n) = Product_{k=1..n} (exp(-8*Zeta'(-1)) * 3^(9*k^2/2 - 3*k + 5/12) * (2*Pi)^(1 - 3*k) * Gamma(k)^2 * Gamma(k + 1/3) * (BarnesG(k) * BarnesG(k + 1/3) * BarnesG(k + 2/3))^3).
a(n) = a(n-1)*A296608(n). - R. J. Mathar, Jul 24 2025

A055746 Product of first n terms of A003046.

Original entry on oeis.org

1, 1, 2, 20, 2800, 16464000, 12778698240000, 4254956888736153600000, 2026001446509988558521630720000000, 4690285643617101997210180025102660272128000000000
Offset: 0

Views

Author

N. J. A. Sloane, Jul 11 2000

Keywords

Crossrefs

Cf. A000108, A003046, A000984, A055462, A255674, A306635.

Programs

  • Maple
    seq(mul(mul(binomial(2*j,j)/(j+1),j=0..k), k=0..n), n=0..9); # Zerinvary Lajos, Sep 21 2007
  • Mathematica
    Table[Product[Product[Binomial[2*j,j]/(j+1),{j,0,k}],{k,0,n}],{n,0,10}] (* Vaclav Kotesovec, Jul 10 2015 *)
Table[Product[2^((k + 1)/2) * Sqrt[BarnesG[2*k]] * Gamma[2*k] / (BarnesG[k] * BarnesG[k + 3] * Gamma[k]^(3/2)), {k, 1, n}], {n, 0, 10}] (* Vaclav Kotesovec, Mar 02 2019 *)

Formula

a(n) ~ c * 2^(n^3/3 + n^2 - n/8 - 71/48) * exp(9*n^2/8 + 5*n/2 - 7/24) * A^(3*n/2 + 4) / (n^(3*n^2/4 + 21*n/8 + 9/4) * Pi^(n^2/4 + 5*n/4 + 27/16)), where A = A074962 = 1.2824271291006226368753425688697917277... is the Glaisher-Kinkelin constant and c = 1.06988379617813356826829257647028132359737354153723273083785714620398... = A255674. - Vaclav Kotesovec, Jul 10 2015
a(n) ~ A^(3*n/2 + 3) * exp(9*n^2/8 + 5*n/2 - 7*Zeta(3)/(32*Pi^2) - 1/4) * 2^(n^3/3 + n^2 - n/8 - 65/48) / (Pi^(n^2/4 + 5*n/4 + 3/2) * n^(3*n^2/4 + 21*n/8 + 9/4)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 02 2019
a(n) = Product_{k=1..n} (2^((k+1)/2) * sqrt(BarnesG(2*k)) * Gamma(2*k) / (BarnesG(k) * BarnesG(k+3) * Gamma(k)^(3/2))). - Vaclav Kotesovec, Mar 02 2019

A066121 Multi-level factorials: triangle with a(n,k)=a(n-1,k-1)*a(n-1,k) but with a(n,1)=n and a(n,n)=1.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 6, 2, 1, 5, 24, 12, 2, 1, 6, 120, 288, 24, 2, 1, 7, 720, 34560, 6912, 48, 2, 1, 8, 5040, 24883200, 238878720, 331776, 96, 2, 1, 9, 40320, 125411328000, 5944066965504000, 79254226206720, 31850496, 192, 2, 1, 10, 362880
Offset: 1

Views

Author

Henry Bottomley, Dec 05 2001

Keywords

Examples

			a(4,2)=a(3,1)*a(3,2)=3*2=6. Rows start 1; 2,1; 3,2,1; 4,6,2,1; ...

Crossrefs

Columns include A000027, A000142, A000178, A055462, A057527, A057528. Right hand side includes A000012, A007395, A007283. Cf. A066119.

A163086 Product of first n terms of A163085.

Original entry on oeis.org

1, 1, 2, 24, 1728, 3732480, 161243136000, 975198486528000000, 412860031256494080000000000, 110116706384632080236544000000000000000, 7401233839469056679744633202278400000000000000000000
Offset: 0

Views

Author

Peter Luschny, Jul 21 2009

Keywords

Crossrefs

Cf. A056040, A163085, A055462, A000178.

Programs

  • Maple
    a := proc(n) local i; mul(A163085(i),i=0..n) end;
  • Mathematica
    b[0] = 1; b[n_] := b[n] = b[n-1] n! / Floor[n/2]!^2;
a[n_] := Product[b[k], {k, 0, n}];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Jul 11 2019 *)
  • Sage
    def A163086(n):
    swing = lambda n: factorial(n)/factorial(n//2)^2
    return mul(swing(i+1)^(n-i) for i in (0..n))
[A163086(i) for i in (0..10)] # Peter Luschny, Sep 18 2012

Formula

a(n) = product_{i=0..n} A056040(i+1)^(n-i). - Peter Luschny, Sep 18 2012

A260610 Suprafactorials: Product of first n hyperfactorials divided by the product of the first n superfactorials.

Original entry on oeis.org

1, 1, 2, 18, 1728, 4320000, 699840000000, 18525482136000000000, 204051433560311070720000000000, 2399547398533110254947300351672320000000000, 77759951835586717141477466390085274435584000000000000000000, 18011357710498321908881994832212360081640749122627567616000000000000000000000000
Offset: 0

Views

Author

Matthew Campbell, Jul 30 2015

Keywords

Examples

			a(3) = (Hyperfactorial(3)/Superfactorial(3)) * (Hyperfactorial(2)/Superfactorial(2)) * (Hyperfactorial(1)/Superfactorial(1)) * (Hyperfactorial(0)/Superfactorial(0)) = ((3^3 * 2^2 * 1^1)/(3! * 2! * 1!)) * ((2^2 * 1^1)/(2!*1!)) * (1^1/1!) * 1 = ((27 * 4)/(6 * 2)) * (4/2) * 1 = (108/12) * (4/2) = 9 * 2 = 18.

Links

Crossrefs

Cf. A000178, A001142, A002109, A055462, A125760, A255269.

Programs

  • Mathematica
    Table[Product[Hyperfactorial[n]/BarnesG[n+2], {n, 0, m}], {m, 0, 12}]
Table[BarnesG[n+2]^(n-1) / Product[BarnesG[k]^3, {k, 1, n + 1}], {n, 0, 12}] (* Vaclav Kotesovec, Nov 19 2023 *)
  • PARI
    a001142(n) = prod(m=1, n, binomial(n, m));
a(n) = prod(k=0, n, a001142(k)); \\ Michel Marcus, Aug 06 2015

Formula

a(n) = A125760(n)/A055462(n).
a(n) = Product_{k=0..n} A001142(k).
a(n) = Product_{k=0..n} hyperfactorial(k)/superfactorial(k).
a(n) = Product_{i=1..n} (Product_{j=1..i} binomial(i,j)). - Pedro Caceres, Apr 13 2019
From Vaclav Kotesovec, Nov 19 2023: (Start)
a(n) = BarnesG(n+2)^(n-1) / Product_{k=1..n+1} BarnesG(k)^3.
a(n) ~ A^(2*n + 5/2) * exp(n^3/6 + 7*n^2/8 + 5*n/6 - 3*zeta(3)/(8*Pi^2) - 1/8) / ((2*Pi)^(n^2/4 + 3*n/4 + 1/2) * n^(n^2/4 + 7*n/12 + 7/24)), where A is the Glaisher-Kinkelin constant A074962. (End)

A365617 Iterated Pochhammer symbol.

Original entry on oeis.org

1, 1, 2, 24, 421200, 13257209623458438290962108800
Offset: 0

Views

Author

Darío Clavijo, Sep 12 2023

Keywords

Links

Crossrefs

Cf. A000142, A000407, A038155, A055462, A112332, A266083.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<0, 0,
      (z-> mul(z+j, j=0..n-1))(a(n-1)))
    end:
seq(a(n), n=0..5);  # Alois P. Heinz, Sep 12 2023
  • Mathematica
    FoldList[Pochhammer, 1, Range[5]] (* Amiram Eldar, Sep 12 2023 *)
  • PARI
    P(x, y) = my(P=1); for (i=0, y-1, P *= x+i); P;
a(n) = my(x=1); n--; for (i=1, n, x = P(x, i+1)); x; \\ Michel Marcus, Sep 13 2023
  • Python
    from gmpy2 import *
from functools import reduce
gamma = lambda n: fac(n - 1)
Pochhammer = lambda z,n: gamma(n + z) // gamma(z)
list_Pochhammer = lambda lst: int(reduce((lambda x, y: Pochhammer(x, y)), lst)) if len(lst) > 0 else 1
print([list_Pochhammer(range(1, n + 1)) for n in range(0, 6)])
  • Python
    from functools import reduce
from sympy import rf
def A365617(n): return reduce(rf,range(1,n+1),1) # Chai Wah Wu, Sep 15 2023

Formula

a(n) = Pochhammer(...(Pochhammer(Pochhammer(1, 2), 3), ...), n).
a(n) = gamma(n + a(n-1)) / gamma(a(n-1)).
a(n) = Product_{j=0..n-1} (j + a(n-1)), a(0) = 1. - Alois P. Heinz, Sep 12 2023
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