A057527 -id:A057527 - 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

A259068 Decimal expansion of zeta'(-3) (the derivative of Riemann's zeta function at -3).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 18 2015

			0.0053785763577743011444169742104138428956644397422955070594470232233245...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15.1 Generalized Glaisher constants, p. 136-137.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's MathWorld, Riemann Zeta Function.
Wikipedia, Riemann Zeta Function
Index entries for constants related to zeta

Cf. A000335, A000391, A000417, A000428, A023872, A057527, A057528, A255050, A255052, A258350, A258351, A258352, A260404.

Join[{0, 0}, RealDigits[Zeta'[-3], 10, 105] // First]

zeta'(-n) = (BernoulliB(n+1)*HarmonicNumber(n))/(n+1) - log(A(n)), where A(n) is the n-th Bendersky constant, that is the n-th generalized Glaisher constant.
zeta'(-3) = -11/720 - log(A(3)), where A(3) is A243263.
Equals -11/720 + (gamma + log(2*Pi))/120 - 3*Zeta'(4)/(4*Pi^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 24 2015

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

Henry Bottomley, Sep 02 2000

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.

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

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 *)

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

Vaclav Kotesovec, Jul 24 2015

nonn

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

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

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

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.

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

Henry Bottomley, Dec 05 2001

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

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

cp's OEIS Frontend

A055462 Superduperfactorials: product of first n superfactorials.

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A259068 Decimal expansion of zeta'(-3) (the derivative of Riemann's zeta function at -3).

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

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

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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.

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