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 1-8 of 8 results.

A055462 Superduperfactorials: product of first n superfactorials.

Original entry on oeis.org

1, 1, 2, 24, 6912, 238878720, 5944066965504000, 745453331864786829312000000, 3769447945987085350501386572267520000000000, 6916686207999802072984424331678589933649915805696000000000000000
Offset: 0

Views

Author

Henry Bottomley, Jun 26 2000

Keywords

Comments

Next term has 92 digits and is too large to display.
Starting with offset 1, a(n) is a 'Matryoshka doll' sequence with alpha=1, the multiplicative counterpart to the additive A000332. The sequence for m with alpha<=m<=L is then computed as Prod_{n=alpha..m}(Prod_{k=alpha..n}(Prod_{i=alpha..k}(i))). - Peter Luschny, Jul 14 2009

Examples

			a(4) = 1!2!3!4!*1!2!3!*1!2!*1! = 288*12*2*1 = 6912.

Links

Crossrefs

Cf. A000142, A000178, A002109, A036740, A255268, A255269, A306635.
Cf. A057527, A057528, A260404.

Programs

  • Magma
    [n eq 0 select 1 else (&*[j^Binomial(n-j+2,2): j in [1..n]]): n in [0..10]]; // G. C. Greubel, Jan 31 2024
  • Maple
    seq(mul(mul(mul(i, i=alpha..k), k=alpha..n), n=alpha..m), m=alpha..10); # Peter Luschny, Jul 14 2009
  • Mathematica
    Table[Product[BarnesG[j], {j, k + 1}], {k, 10}] (* Jan Mangaldan, Mar 21 2013 *)
Table[Round[Exp[(n+2)*(n+3)*(2*n+5)/8] * Exp[PolyGamma[-3, n+3]] * BarnesG[n+3]^(n+3/2) / (Glaisher^(n+3) * (2*Pi)^((n+3)^2/4) * Gamma[n+3]^((n+2)^2/2))], {n, 0, 10}] (* Vaclav Kotesovec, Feb 20 2015 after Jan Mangaldan *)
Nest[FoldList[Times,#]&,Range[0,15]!,2]  (* Harvey P. Dale, Jul 14 2023 *)
  • PARI
    a(n)=my(t=1);prod(k=2,n,t*=k!) \\ Charles R Greathouse IV, Jul 28 2011
  • SageMath
    [product(j^binomial(n-j+2,2) for j in range(1,n+1)) for n in range(11)] # G. C. Greubel, Jan 31 2024

Formula

a(n) = a(n-1)*A000178(n) = Product_{i=1..n} (i!)^(n-i+1) = Product_{i=1..n} i^((n-i+1)*(n-i+2)/2).
log a(n) = (1/6) n^3 log n - (11/36) n^3 + O(n^2 log n). - Charles R Greathouse IV, Jan 13 2012
a(n) = exp((6 + 13 n + 9 n^2 + 2 n^3 - 8*(n + 2)*log(A)-2*(n + 2)^2*log(2*Pi) + 4*(2 n + 1)*logG(n + 2) - 4*(n + 1)^2*logGamma(n + 2) + 8*psi(-3, n + 2))/8) where A is the Glaisher-Kinkelin constant, logG(z) is the logarithm of the Barnes G function (A000178), and psi(-3, z) is a polygamma function of negative order (i.e., the second iterated integral of logGamma(z)). - Jan Mangaldan, Mar 21 2013
a(n) ~ exp(Zeta(3)/(8*Pi^2) - (2*n+3)*(11*n^2 + 24*n - 3)/72) * n^((2*n+3)*(2*n^2 + 6*n + 3)/24) * (2*Pi)^((n+1)*(n+2)/4) / A^(n+3/2), where A = A074962 = 1.28242712910062263687... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.2020569031595942853997... . - Vaclav Kotesovec, Feb 20 2015

Extensions

a(9) from N. J. A. Sloane, Dec 15 2008

A255269 a(n) = Product_{k=1..n} k!^k.

Original entry on oeis.org

1, 4, 864, 286654464, 7132880358604800000, 993710590042385551668019200000000000, 82086865668400428790437436119503664712777728000000000000000000
Offset: 1

Views

Author

Vaclav Kotesovec, Feb 20 2015

Keywords

Links

Crossrefs

Cf. A000178, A002109, A036740, A055462, A255268.

Programs

  • Mathematica
    Table[Product[k!^k,{k,1,n}],{n,1,10}]
FoldList[Times,Table[(k!)^k,{k,10}]] (* Harvey P. Dale, Aug 16 2021 *)

Formula

a(n) = A255268(n) / A055462(n-1).
a(n) ~ sqrt(A) * exp((3 - 45*n^2 - 32*n^3 - 9*Zeta(3)/Pi^2)/72) * n^((8*n^3 + 18*n^2 + 10*n + 1)/24) * (2*Pi)^(n*(n+1)/4), where A = A074962 = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.2020569031595942853997... .

A255322 a(n) = Product_{k=0..n} (k^2)!.

Original entry on oeis.org

1, 1, 24, 8709120, 182219087869378560000, 2826438545846116156142906806150103040000000000, 1051416277636507481568264360276689674557030810000137484550133942059008000000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 21 2015

Keywords

Comments

Partial products of A088020. - Michel Marcus, Jul 06 2019

Links

Crossrefs

Cf. A000178, A255358, A255359, A255360.
Cf. A002109, A051675, A255321, A255323, A255344.
Cf. A000142, A255268, A255269, A255504.
Cf. A272094, A372240, A372241.

Programs

  • Mathematica
    Table[Product[(k^2)!, {k, 0, n}], {n, 0, 10}]
FoldList[Times,(Range[0,6]^2)!] (* Harvey P. Dale, Jan 30 2022 *)
Table[(n^2)!^(n+1) / Product[j^(Ceiling[Sqrt[j]]), {j, 1, n^2}], {n, 0, 6}] (* Vaclav Kotesovec, Apr 23 2024 *)
Table[(n^2)!^n * (n!)^2 / Product[j^(Floor[Sqrt[j]]), {j, 1, n^2}], {n, 0, 6}] (* Vaclav Kotesovec, Apr 23 2024 *)
  • PARI
    {a(n) = prod(k=1, n, (k^2)!)} \\ Seiichi Manyama, Jul 06 2019

Formula

a(n) ~ c * n^((2*n + 1)*(2*n^2 + 2*n + 3)/6) * (2*Pi)^(n/2) / exp(5*n^3/9 + n^2/2 + n), where c = A255504 = 3.048330306522348566911920417337613015885313475... .
From Vaclav Kotesovec, Apr 23 2024: (Start)
a(n) = Product_{j=1..n^2} j^(n - ceiling(sqrt(j)) + 1).
a(n) = (n^2)!^n * (n!)^2 / Product_{j=1..n^2} j^(floor(sqrt(j))). (End)

A255358 Product_{k=0..n} (k^3)!.

Original entry on oeis.org

1, 1, 40320, 439039216240867959122165760000000
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 21 2015

Keywords

Comments

The next term a(4) has 122 digits.

Links

Crossrefs

Cf. A000178, A255322, A255359, A255360.
Cf. A002109, A051675, A255321, A255323, A255344.
Cf. A000142, A255268, A255269, A255511.

Programs

  • Mathematica
    Table[Product[(k^3)!, {k, 0, n}], {n, 0, 6}]
Table[Product[j^(n - Ceiling[j^(1/3)] + 1), {j, 1, n^3}], {n, 0, 6}] (* Vaclav Kotesovec, Apr 25 2024 *)

Formula

a(n) ~ c * n^(29/40 + 3*n/2 + 3*n^2/4 + 3*n^3/2 + 3*n^4/4) * (2*Pi)^(n/2) / exp(n*(n+2)*(12 - 6*n + 7*n^2)/16), where c = A255511 = 4.113740552015338123052453340090368136...
a(n) = Product_{j=1..n^3} j^(n - ceiling(j^(1/3)) + 1). - Vaclav Kotesovec, Apr 25 2024

A255359 a(n) = Product_{k=0..n} (k^4)!.

Original entry on oeis.org

1, 1, 20922789888000
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 21 2015

Keywords

Comments

The next term a(3) has 135 digits.

Links

Crossrefs

Cf. A000178, A255322, A255358, A255360.
Cf. A002109, A051675, A255321, A255323, A255344.
Cf. A000142, A255268, A255269, A255438.

Programs

  • Mathematica
    Table[Product[(k^4)!, {k, 0, n}], {n, 0, 5}]
Table[Product[j^(n - Ceiling[j^(1/4)] + 1), {j, 1, n^4}], {n, 0, 5}] (* Vaclav Kotesovec, Apr 25 2024 *)

Formula

a(n) ~ c * n^(1 + 28*n/15 + 4*n^3/3 + 2*n^4 + 4*n^5/5) * (2*Pi)^(n/2) / exp(19*n/9 + n^4/2 + 9*n^5/25), where c = A255438 = 6.644987918706354049483118... .
a(n) = Product_{j=1..n^4} j^(n - ceiling(j^(1/4)) + 1). - Vaclav Kotesovec, Apr 25 2024

A255360 Product_{k=0..n} (k^5)!.

Original entry on oeis.org

1, 1, 263130836933693530167218012160000000
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 21 2015

Keywords

Comments

The next term a(3) has 512 digits.
In general (for m>1), product_{k=0..n} (k^m)! ~ c(m) * (2*Pi)^(n/2) * n^(m*(1/4 + n/2 + B(m+1)/(m+1) + (sum_{j=1..n} j^m) )) * exp(-m*n/2 - m*n^(m+1)/(m+1)^2 - (sum_{j=1..n} j^m) + m * (sum_{j=1..m-1} 1/(j+1) * B(j+1) * binomial(m, j) * n^(m-j) * (sum_{i=0..j-1} 1/(m-i)) )), where c(m) is a constant and B(n) is the Bernoulli number A027641(n)/A027642(n).

Links

Crossrefs

Cf. A000178, A255322, A255358, A255359.
Cf. A002109, A051675, A255321, A255323, A255344.
Cf. A000142, A255268, A255269, A255439.

Programs

  • Mathematica
    Table[Product[(k^5)!, {k, 0, n}], {n, 0, 4}]
Table[Product[j^(n - Ceiling[j^(1/5)] + 1), {j, 1, n^5}], {n, 0, 4}] (* Vaclav Kotesovec, Apr 25 2024 *)

Formula

a(n) ~ c * n^(80/63 + 5*n/2 - 5*n^2/12 + 25*n^4/12 + 5*n^5/2 + (5*n^6)/6) * (2*Pi)^(n/2) / exp(5*n/2 + 35*n^2/144 + n^5/2 + 11*n^6/36), where c = A255439 = 11.354954749729782312106... .
a(n) = Product_{j=1..n^5} j^(n - ceiling(j^(1/5)) + 1). - Vaclav Kotesovec, Apr 25 2024

A362288 a(n) = Product_{k=0..n} binomial(n,k)^k.

Original entry on oeis.org

1, 1, 2, 27, 9216, 312500000, 4251528000000000, 95432797246104853383515625, 14719075154533285649961930052505436160000, 65577306173662530591576256095315195684570038194755952705536
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 14 2023

Keywords

Links

Crossrefs

Cf. A001142, A067055, A167008, A255268, A272093.

Programs

  • Mathematica
    Table[Product[Binomial[n, k]^k, {k, 0, n}], {n, 0, 10}]
Table[(n!)^(n*(n+1)/2) / BarnesG[n+2]^n, {n, 0, 10}]
  • PARI
    a(n) = prod(k=0, n, binomial(n,k)^k); \\ Michel Marcus, Apr 14 2023

Formula

a(n) = Product_{k=0..n} n!^k / k!^n.
a(n) = A067055(n) / A255268(n).
a(n) ~ A^n * exp((6*n^3 + 12*n^2 - n - 1)/24) / ((2*Pi)^(n*(n+1)/4) * n^(n*(3*n+2)/12)), where A is the Glaisher-Kinkelin constant A074962.

A255403 Product_{k=1..n} (k^k)!.

Original entry on oeis.org

1, 24, 261332866810040451858432000000
Offset: 1

Views

Author

Vaclav Kotesovec, Feb 22 2015

Keywords

Comments

The next term (a(4)) has 537 digits.

Links

Crossrefs

Cf. A000178, A255322, A255358, A255359, A255360.
Cf. A002109, A051675, A255321, A255323, A255344.
Cf. A000142, A255268, A255269.

Programs

  • Mathematica
    Table[Product[(k^k)!, {k, 1, n}], {n, 1, 4}]
Showing 1-8 of 8 results.

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

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