A255269 -id:A255269 - cp's OEIS frontend

A243262 Decimal expansion of the generalized Glaisher-Kinkelin constant A(2).

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

Offset: 1

Jean-François Alcover, Jun 02 2014

Also known as the second Bendersky constant.

This is likely the same as the constant B considered in section 3 of the Choi and Srivastava link. - R. J. Mathar, Oct 03 2016

			1.03091675219739211419331309646694229...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.

G. C. Greubel, Table of n, a(n) for n = 1..2002
J. Choi and H. M. Srivastava, Certain classes of series involving the zeta function, J. Math. Annal. Applic. 231 (1999) 91-117.
K. Kimoto, N. Kurokawa, C. Sonoki, M. Wakayama, Some examples of generalized zeta regularized products, Kodai Math. J. 27 (2004), 321-335.
Tobias Kyrion, A closed-form expression for zeta(3), arXiv:2008.05573 [math.GM], 2020.
Eric Weisstein's MathWorld, Glaisher-Kinkelin Constant

Cf. A051675, A055462, A240966, A255269.
Cf. A019727, A074962, A243263, A243264, A243265, A266553, A266554, A266555, A266556, A266557, A266558, A266559, A260662, A266560, A266562, A266563, A266564, A266565, A266566, A266567.
Cf. A002117 (zeta(3)).

RealDigits[Exp[Zeta[3]/(4*Pi^2)], 10, 99] // First
RealDigits[Exp[N[(BernoulliB[2]/4)*(Zeta[3]/Zeta[2]), 200]]]//First (* G. C. Greubel, Dec 31 2015 *)

exp(zeta(3)/(4*Pi^2)) \\ Felix Fröhlich, Jun 27 2019

A(k) = exp(B(k+1)/(k+1)*H(k)-zeta'(-k)), where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.
A(0) = sqrt(2*Pi) (A019727),
A(1) = A = Glaisher-Kinkelin constant (A074962),
A(2) = exp(-zeta'(-2)) = exp(zeta(3)/(4*Pi^2)).
Equals exp(-A240966). - Vaclav Kotesovec, Feb 22 2015

A055462 Superduperfactorials: product of first n superfactorials.

Original entry on oeis.org

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

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

Original entry on oeis.org

1, 1, 24, 8709120, 182219087869378560000, 2826438545846116156142906806150103040000000000, 1051416277636507481568264360276689674557030810000137484550133942059008000000000000000000

Offset: 0

Vaclav Kotesovec, Feb 21 2015

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 0..12

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

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 *)

{a(n) = prod(k=1, n, (k^2)!)} \\ Seiichi Manyama, Jul 06 2019

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)

A255268 a(n) = Product_{k=1..n} k!^n.

Original entry on oeis.org

1, 4, 1728, 6879707136, 49302469038676377600000, 237376313799769806328950291431424000000000000, 487929826521303413461947888047619993419888153407795494912000000000000000000000

Offset: 1

Vaclav Kotesovec, Feb 20 2015

nonn

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

Cf. A000178, A055462, A074962, A255269.

Table[Product[k!,{k,1,n}]^n,{n,1,10}]
Table[BarnesG[n+2]^n, {n, 1, 10}]

a(n) = A000178(n)^n.

a(n) ~ exp(1/12 + n/12 - n^2 - 3*n^3/4) * n^(5*n/12 + n^2 + n^3/2) * 2^(n/2 + n^2/2) * Pi^(n/2 + n^2/2) / A^n, where A = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant (see A074962).

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

Original entry on oeis.org

1, 1, 40320, 439039216240867959122165760000000

Offset: 0

Vaclav Kotesovec, Feb 21 2015

nonn

The next term a(4) has 122 digits.

Vaclav Kotesovec, Table of n, a(n) for n = 0..6

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

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 *)

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

Vaclav Kotesovec, Feb 21 2015

nonn

The next term a(3) has 135 digits.

Vaclav Kotesovec, Table of n, a(n) for n = 0..4

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

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 *)

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

Vaclav Kotesovec, Feb 21 2015

nonn

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).

Vaclav Kotesovec, Table of n, a(n) for n = 0..3

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

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 *)

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

A125760 a(n) = Product_{k=1..n} A002109(k).

Original entry on oeis.org

1, 1, 4, 432, 11943936, 1031956070400000, 4159895825138319360000000000, 13809882382682787973867537170432000000000000000, 769161257109634779902443718589603914508004789479014400000000000000000000, 16596916396875768196482032091931000424134701157007816971266990744831779993781534720000000000000000000000000

Offset: 0

N. J. A. Sloane, based on a suggestion from J. M. Bergot, Feb 06 2007

nonn

N. J. A. Sloane, Table of n, a(n) for n = 0..18

Cf. A002109, A255269.

seq(mul(mul(mul(k,j=1..k), k=1..m), m=1..n), n=0..9); # Zerinvary Lajos, Jun 01 2007

Table[Product[Gamma[1 + k]^k/BarnesG[1 + k], {k, 1, n}], {n, 0, 10}] (* Vaclav Kotesovec, Nov 19 2023 *)
Table[BarnesG[n + 2]^n/Product[BarnesG[k]^2, {k, 1, n + 1}], {n, 0, 10}] (* Vaclav Kotesovec, Nov 19 2023 *)

From Vaclav Kotesovec, Nov 19 2023: (Start)
a(n) = BarnesG(n+2)^n / Product_{k=1..n+1} BarnesG(k)^2.
a(n) ~ A^(n+1) * n^(n^3/6 + n^2/2 + 5*n/12 + 1/12) / exp(5*n^3/36 + n^2/4 + n/12 + zeta(3)/(4*Pi^2)), where A is the Glaisher-Kinkelin constant A074962. (End)

A367492 a(n) = Product_{k=0..n} (k+1)!^k.

Original entry on oeis.org

1, 2, 72, 995328, 206391214080000, 39934999921327865856000000000, 654541076770994951831125144608178176000000000000000, 113391518341540395635327816456127297986876881699306137641287680000000000000000000000

Offset: 0

Vaclav Kotesovec, Nov 20 2023

nonn

G. C. Greubel, Table of n, a(n) for n = 0..15

Cf. A203421, A255269.

[(&*[Factorial(k+1)^k: k in [0..n]]): n in [0..15]]; // G. C. Greubel, Feb 18 2024

Table[Product[(k+1)!^k, {k, 0, n}], {n, 0, 10}]

[product(factorial(k+1)^k for k in range(n+1)) for n in range(16)] # G. C. Greubel, Feb 18 2024

a(n) ~ A^(3/2) * n^(n^3/3 + 5*n^2/4 + 11*n/12 - 3/8) * (2*Pi)^(n^2/4 + n/4 - 1/2) / exp(4*n^3/9 + 7*n^2/8 - n + zeta(3)/(8*Pi^2) - 25/24), where A is the Glaisher-Kinkelin constant A074962.

a(n) = (n+1)^n * abs(A203421(n)) * A255269(n).

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

Original entry on oeis.org

1, 24, 261332866810040451858432000000

Offset: 1

Vaclav Kotesovec, Feb 22 2015

nonn

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

Vaclav Kotesovec, Table of n, a(n) for n = 1..4

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

Table[Product[(k^k)!, {k, 1, n}], {n, 1, 4}]

cp's OEIS Frontend

A243262 Decimal expansion of the generalized Glaisher-Kinkelin constant A(2).

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A055462 Superduperfactorials: product of first n superfactorials.

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

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A255268 a(n) = Product_{k=1..n} k!^n.

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

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

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

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