A306635 -id:A306635 - cp's OEIS frontend

A055462 Superduperfactorials: product of first n superfactorials.

1, 1, 2, 24, 6912, 238878720, 5944066965504000, 745453331864786829312000000, 3769447945987085350501386572267520000000000, 6916686207999802072984424331678589933649915805696000000000000000

Offset: 0

Henry Bottomley, Jun 26 2000

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

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

Miles Seventh, Table of n, a(n) for n = 0..20
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.

Cf. A000142, A000178, A002109, A036740, A255268, A255269, A306635.

Cf. A057527, A057528, A260404.

[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

seq(mul(mul(mul(i, i=alpha..k), k=alpha..n), n=alpha..m), m=alpha..10); # Peter Luschny, Jul 14 2009

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 *)

a(n)=my(t=1);prod(k=2,n,t*=k!) \\ Charles R Greathouse IV, Jul 28 2011

[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

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

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

A168467 a(n) = Product_{k=0..n} ((2k+2)(2*k+3))^(n-k).

Original entry on oeis.org

1, 6, 720, 3628800, 1316818944000, 52563198423859200000, 327312129899898454671360000000, 428017682605583614976547335700480000000000, 152240508705590071980086429193304853792686080000000000000

Offset: 0

Paul Barry, Nov 26 2009

Hankel transform of A000698(n+1).
The sequence 1,1,6,720,... with general term Product_{k=0..n, ((2k+1)(2k+0^k))^(n-k)} is the Hankel transform of A112934. - Paul Barry, Dec 04 2009
a(n) is also the determinant of the n X n matrix M(i,j) = i^(2*j)*sinh(2*j*arccsch(i))/(2*sqrt(i^2+1)), with i and j from 1 to n, which is the same matrix generated by sequences of length n by the linear recurrences with kernel { 2*(k^2 + z), -k^4 }, and initial conditions { 1, 2*(k^2 + z) }, with k from 1 to n, and z = 2. Regardless of the value of z, for every n, the determinant of the n X n matrix of polynomials generated gives always a(n) as result. - Federico Provvedi, Feb 01 2021

			From _Federico Provvedi_, Apr 01 2021: (Start)
From both formulas in the comment above and in particular with z=2 from the linear recurrences, the determinant of the 5 X 5 matrix: ( (1,6,35,204,1189), (1,12,128,1344,14080),(1,22,403,7084,123205), (1,36,1040,28224,749824), (1,54,2291,89964,3426181) ) = 1316818944000 = a(5).
For a generic z, the determinant doesn't change as shown in this example, where the determinant of the 3 X 3 square matrix:
( ( 1, 2*(z+1), (2*z + 1)*(2*z+3)  ),
  ( 1, 2*(z+4), 4*(z+6)*(z+2)      ),
  ( 1, 2*(z+9), (2*z + 9)(2*z + 27)) ) = 720 = a(3). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..28

Cf. A000178, A000698, A098694, A112934, A306635, A001109, A099157, A246645, A202768, A000142.

Table[2^(n^2 + 2*n + 23/24) Glaisher^(3/2) Pi^(-n/2 - 3/4) BarnesG[n + 2] BarnesG[n + 5/2]/E^(1/8), {n, 0, 10}] (* Vladimir Reshetnikov, Sep 06 2016 *)
Table[Product[((2k+2)(2k+3))^(n-k),{k,0,n}],{n,0,10}] (* Harvey P. Dale, Dec 26 2019 *)
Table[Det@Table[LinearRecurrence[{2*k^2,-k^4},{1, 2*k^2},n], {k, 1, n}], {n,1,20}] (* Federico Provvedi, Feb 01 2021 *)
Det@Expand@Array[(#1^(2 #2))/(4 Sqrt[1 + #1^2])((Sqrt[1+1/#1^2]+1/#1)^(2 #2)-(Sqrt[1+1/#1^2]-1/#1)^(2 #2))&,{#,#}]&/@Range[20] (* Federico Provvedi, Apr 01 2021 *)

from math import prod
def A168467(n): return prod(((m:=k+1<<1)*(m+1))**(n-k) for k in range(1,n+1))*3**n<Chai Wah Wu, Nov 26 2023

G.f.: Q(0)/(2*x) -1/x, where Q(k) = 1  + 1/(1 -(2*k+1)!*x/((2*k+1)!*x + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 17 2013
a(n) = Product_{k=1..n} (2*k+1)!. - Vladimir Reshetnikov, Sep 06 2016
a(n) ~ A^(-1/2) * 2^(n^2 + 3*n + 53/24) * exp((-3/2)*n^2 + (-5/2)*n + 1/24) * n^(n^2 + (5/2)*n + 35/24) * Pi^((n+1)/2), where A = A074962 is the Glaisher-Kinkelin constant. - Vladimir Reshetnikov, Sep 06 2016
a(n) = A000178(2*n + 1) / A098694(n). - Vaclav Kotesovec, Oct 28 2017
a(n) = A202768(n)*A000142(n). - Federico Provvedi, Feb 01 2021
For n > 0, a(n) = n * (2*n+1) * sqrt(BarnesG(2*n)) * Gamma(2*n)^2 / (sqrt(Gamma(n)) * 2^((n-3)/2)). - Vaclav Kotesovec, Nov 27 2024

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

Original entry on oeis.org

1, 288, 36118462464000, 240498631970530185123135341199360000000000

Offset: 1

Vaclav Kotesovec, Mar 03 2019

nonn

Next term is too long to be included.

Eric Weisstein's World of Mathematics, Barnes G-Function.
Wikipedia, Barnes G-function

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

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 *)

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

A296607 a(n) = BarnesG(2*n).

Original entry on oeis.org

0, 1, 2, 288, 24883200, 5056584744960000, 6658606584104736522240000000, 127313963299399416749559771247411200000000000, 69113789582492712943486800506462734562847413501952000000000000000

Offset: 0

Vaclav Kotesovec, Dec 16 2017

nonn

Eric Weisstein's World of Mathematics, Barnes G-Function.
Wikipedia, Barnes G-function

Cf. A000178, A098694, A296591, A296608, A306635.

Table[BarnesG[2*n], {n, 0, 10}]
Table[Glaisher^3 * E^(-1/4) * 2^(2*n^2 - 3*n + 11/12) * Pi^(1/2 - n) * BarnesG[n] * BarnesG[n + 1/2]^2 * BarnesG[n+1], {n, 0, 10}]

a(n) = A^3 * exp(-1/4) * 2^(2*n^2 - 3*n + 11/12) * Pi^(1/2 - n) * BarnesG(n) * BarnesG(n + 1/2)^2 * BarnesG(n+1), where A is the Glaisher-Kinkelin constant A074962.
a(n) ~ 2^(2*n^2 - n - 1/12) * exp(1/12 + 2*n - 3*n^2) * n^(2*n^2 - 2*n + 5/12) * Pi^(n - 1/2) / A, where A is the Glaisher-Kinkelin constant A074962.
a(n) = A000178(2*n-2), n>0. - 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

N. J. A. Sloane, Jul 11 2000

nonn

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

seq(mul(mul(binomial(2*j,j)/(j+1),j=0..k), k=0..n), n=0..9); # Zerinvary Lajos, Sep 21 2007

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 *)

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

A255674 Decimal expansion of a constant related to the Barnes G-function.

Original entry on oeis.org

1, 0, 6, 9, 8, 8, 3, 7, 9, 6, 1, 7, 8, 1, 3, 3, 5, 6, 8, 2, 6, 8, 2, 9, 2, 5, 7, 6, 4, 7, 0, 2, 8, 1, 3, 2, 3, 5, 9, 7, 3, 7, 3, 5, 4, 1, 5, 3, 7, 2, 3, 2, 7, 3, 0, 8, 3, 7, 8, 5, 7, 1, 4, 6, 2, 0, 3, 9, 8, 6, 3, 0, 9, 0, 7, 2, 2, 3, 1, 3, 3, 7, 7, 2, 7, 0, 8, 5, 9, 8, 9, 9, 3, 0, 5, 9, 6, 8, 0, 3, 5, 7, 0, 5, 4

Offset: 1

Vaclav Kotesovec, Jul 10 2015

			1.06988379617813356826829257647028132359737354153723273083785714620398...

Eric Weisstein's World of Mathematics, Barnes G-Function
Wikipedia, Barnes G-function

Cf. A055746, A306635.

(* The iteration cycle: *) $MaxExtraPrecision = 1000; funs[n_]:=Product[BarnesG[j+1/2] / BarnesG[j], {j, 1, n}] / (Glaisher^(1/2) * n^(n^2/4 - n/8 - 1/24) * (2*Pi)^(n/4 - 3/16) / E^(n*(3*n-1)/8)); Do[Print[N[Sum[(-1)^(m + j)*funs[j*Floor[200/m]]*(j^(m - 1)/(j - 1)!/(m - j)!), {j, 1, m}], 120]], {m, 10, 150, 10}]
RealDigits[2^(1/8) * Pi^(3/16) * E^(1/24 - 7*Zeta[3]/(32*Pi^2)) / Glaisher, 10, 120][[1]] (* Vaclav Kotesovec, Mar 02 2019 *)

Equals limit n->infinity (Product_{j = 1..n} BarnesG(j + 1/2) / BarnesG(j)) / (A^(1/2) * n^(n^2/4 - n/8 - 1/24) * (2*Pi)^(n/4 - 3/16) / exp(n*(3*n-1)/8)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant.
Equals limit n->infinity A055746(n) / (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))).
From Vaclav Kotesovec, Mar 02 2019: (Start)
Equals 2^(1/8) * Pi^(3/16) * exp(1/24 - 7*Zeta(3)/(32*Pi^2)) / A, where A is the Glaisher-Kinkelin constant A074962.
Equals exp(-1/24 - 7*Zeta(3)/(32*Pi^2) + Zeta'(-1) + log(2)/8 + 3*log(Pi)/16).
(End)

A324992 Decimal expansion of zeta'(-1, 1/2).

Original entry on oeis.org

0, 5, 3, 8, 2, 9, 4, 3, 9, 3, 2, 6, 8, 9, 4, 4, 1, 0, 0, 4, 7, 9, 0, 8, 4, 9, 1, 7, 2, 7, 2, 9, 9, 6, 3, 1, 0, 4, 5, 5, 3, 9, 0, 1, 7, 9, 0, 2, 5, 9, 0, 2, 5, 6, 2, 4, 4, 8, 9, 9, 4, 8, 6, 1, 1, 6, 4, 5, 5, 1, 1, 5, 5, 8, 4, 5, 5, 1, 3, 0, 6, 5, 6, 2, 8, 5, 1, 5, 7, 8, 2, 0, 8, 0, 7, 0, 2, 6, 5, 7, 8, 8, 2, 7, 1, 8

Offset: 0

Vaclav Kotesovec, Mar 23 2019

			0.053829439326894410047908491727299631045539017902590256244899486116455...

J. Miller and V. Adamchik, Derivatives of the Hurwitz Zeta Function for Rational Arguments, Journal of Computational and Applied Mathematics 100 (1998) 201-206. [contains a large number of typos]
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function, formula 22.

Cf. A074962, A084448, A240966, A261829, A306635.

evalf(Zeta(1,-1,1/2), 120);
evalf(-log(2)/24 - Zeta(1,-1)/2, 120);

RealDigits[Derivative[1, 0][Zeta][-1, 1/2], 10, 120][[1]]
N[With[{k=1}, -BernoulliB[2*k] * Log[2] / 4^k / k - (2^(2*k - 1) - 1) * Zeta'[1 - 2*k] / 2^(2*k - 1)], 120]

zetahurwitz'(-1, 1/2) \\ Michel Marcus, Mar 24 2019

Equals -log(2)/24 - Zeta'(-1)/2 = A261829 - log(2)/24.
Equals -1/24 - log(2)/24 + log(A)/2, where A is the Glaisher-Kinkelin constant A074962.
Equals (log(Pi) - 1 + gamma)/24 - Zeta'(2)/(4*Pi^2), where gamma is the Euler-Mascheroni constant A001620.

A365266 a(n) = Product_{k=1..n} Gamma(6*k).

Original entry on oeis.org

1, 120, 4790016000, 1703748471578689536000000, 44045334006101976766560297729172439040000000000, 389438360216723307909581902233109465138002465491175688781168640000000000000000

Offset: 0

Vaclav Kotesovec, Sep 01 2023

nonn

Cf. A000178, A168467, A294319, A294322, A294326.

Cf. A055462, A306635, A306651.

Table[Product[Gamma[6*k], {k, 1, n}], {n, 0, 10}]
Table[Product[(6*k-1)!, {k, 1, n}], {n, 0, 10}]

a(n) = A^(35/6) * exp(-35/72) * Gamma(1/3)^(5/3) * 2^(-125/72 + 3*n^2) * 3^(47/72 + 5*n/2 + 3*n^2) * Pi^(-25/12 - 5*n/2) * BarnesG(1 + n) * BarnesG(7/6 + n) * BarnesG(4/3 + n) * BarnesG(3/2 + n) * BarnesG(5/3 + n) * BarnesG(11/6 + n), where A = A074962 is the Glaisher-Kinkelin constant.

a(n) ~ A^(-1/6) * Gamma(1/3)^(5/3) * 2^(-35/72 + 3*n + 3*n^2) * 3^(47/72 + 5*n/2 + 3*n^2) * exp(1/72 - 5*n/2 - 9*n^2/2) * n^(19/72 + 5*n/2 + 3*n^2) * Pi^(-5/6 + n/2), where A = A074962 is the Glaisher-Kinkelin constant.

cp's OEIS Frontend

A055462 Superduperfactorials: product of first n superfactorials.

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A168467 a(n) = Product_{k=0..n} ((2*k+2)*(2*k+3))^(n-k).

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

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A296607 a(n) = BarnesG(2*n).

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A055746 Product of first n terms of A003046.

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A255674 Decimal expansion of a constant related to the Barnes G-function.

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A324992 Decimal expansion of zeta'(-1, 1/2).

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A365266 a(n) = Product_{k=1..n} Gamma(6*k).

A168467 a(n) = Product_{k=0..n} ((2k+2)(2*k+3))^(n-k).