A074962 -id:A074962 - 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

A035528 Euler transform of A027656(n-1).

Original entry on oeis.org

0, 1, 1, 3, 3, 6, 9, 13, 19, 28, 42, 57, 84, 115, 164, 227, 313, 429, 588, 799, 1079, 1461, 1952, 2617, 3480, 4627, 6111, 8072, 10604, 13905, 18181, 23701, 30828, 39990, 51763, 66822, 86124, 110687, 142039, 181841, 232409, 296401, 377419, 479635, 608558, 770818

Offset: 0

Christian G. Bower

nonn

Also the weigh transform of A003602. - John Keith, Nov 17 2021

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.

Cf. A000219, A003602, A052847, A263150, A263352, A262876, A263136, A263141.

nmax = 50; CoefficientList[Series[-1 + Product[1/(1 - x^(2*k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2015 *)
nmax = 100; Flatten[{0, Rest[CoefficientList[Series[E^Sum[1/j*x^j/(1 - x^(2*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]]}] (* Vaclav Kotesovec, Oct 10 2015 *)

a(n) ~ A^(1/2) * Zeta(3)^(11/72) * exp(-1/24 - Pi^4/(1728*Zeta(3)) + Pi^2 * n^(1/3)/(3*2^(8/3)*Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2^(4/3)) / (sqrt(3*Pi) * 2^(71/72) * n^(47/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Oct 02 2015

A007685 a(n) = Product_{k=1..n} binomial(2*k,k).

Original entry on oeis.org

1, 2, 12, 240, 16800, 4233600, 3911846400, 13425456844800, 172785629592576000, 8400837310791045120000, 1552105098192510332190720000, 1094904603628138948657963991040000, 2960792853328653706847125274154762240000, 30794022150329995743434211126374020153344000000

Offset: 0

N. J. A. Sloane

H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..50
H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83. (Annotated scanned copy)
Bernd C. Kellner, Asymptotic products of binomial and multinomial coefficients revisited, Integers 24 (2024), Article #A59, 10 pp.; arXiv:2312.11369 [math.CO], 2023.

Cf. A000984, A001142, A112332, A203471, A268196, A296589, A296590, A338550.

[seq(mul(binomial(2*k,k),k=1..n),n=0..16)];

Table[Product[Binomial[2*k, k], {k, 1, n}], {n, 0, 50}] (* G. C. Greubel, Feb 02 2017 *)

a(n) = prod(k=1,n, binomial(2*k, k)); \\ Michel Marcus, Sep 18 2015

a(0) = 1, a(n) = (2^(2*n)*a(n - 1)*Gamma(n + 1/2))/(sqrt(Pi)*Gamma(n + 1)). - Ilya Gutkovskiy, Sep 18 2015
a(n) = (2^(n^2 + n - 1/24)*A^(3/2)*Pi^(-n/2 - 1/4)*BarnesG(n + 3/2))/(e^(1/8)*BarnesG(n + 2)), where A is the Glaisher-Kinkelin constant (A074962), BarnesG is the Barnes G-function. - Ilya Gutkovskiy, Sep 18 2015
a(n) ~ A^(3/2) * 2^(n^2 + n - 7/24) * exp(n/2 - 1/8) / (Pi^((n+1)/2) * n^(n/2 + 3/8)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Apr 16 2016
For n>0, a(n) = 2^((n+1)/2) * sqrt(BarnesG(2*n)) * Gamma(2*n) / (n * BarnesG(n)^2 * Gamma(n)^(7/2)). - Vaclav Kotesovec, Apr 20 2024
Product_{1 <= j <= i <= n}  (i + j)/(i - j + 1). - Peter Bala, Oct 25 2024

A007875 Number of ways of writing n as p*q, with p <= q, gcd(p, q) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 4, 1, 2, 2, 4, 1, 2, 1, 2, 2, 2, 2, 4, 1, 2, 1, 2, 1, 4, 2, 2, 2, 2, 1, 4

Offset: 1

Victor Ufnarovski

a(n), n >= 2, is the number of divisor products in the numerator as well as denominator of the unique representation of n in terms of divisor products. See the W. Lang link under A007955, where a(n)=l(n) in Table 1. - Wolfdieter Lang, Feb 08 2011
Record values are the binary powers, occurring at primorial positions except at 2: a(A002110(0))=A000079(0), a(A002110(n+1))=A000079(n) for n > 0. - Reinhard Zumkeller, Aug 24 2011
For n > 1: a(n) = (A000005(n) - A048105(n)) / 2; number of ones in row n of triangle in A225817. - Reinhard Zumkeller, Jul 30 2013

T. D. Noe, Table of n, a(n) for n = 1..1000
Larry Bates and Peter Gibson, A geometry where everything is better than nice, arXiv:1603.06622 [math.DG], (21-March-2016); see page 2.
Vaclav Kotesovec, Graph - the asymptotic ratio.

Cf. A000005, A000010, A000079, A001221, A001620, A002110, A007955, A034444, A048105, A074962.

a007875 = length . filter (> 0) . a225817_row
-- Reinhard Zumkeller, Jul 30 2013, Aug 24 2011

A007875 := proc(n)
    if n = 1 then
        1;
    else
        2^(A001221(n)-1) ;
    end if;
end proc: # R. J. Mathar, May 28 2016

a[n_] := With[{r = Reduce[1 <= p <= q <= n && n == p*q && GCD[p, q] == 1, {p, q}, Integers]}, If[Head[r] === And, 1, Length[r]]]; Table[a[n], {n, 1, 90}] (* Jean-François Alcover, Nov 02 2011 *)
a[n_] := EulerPhi[2^PrimeNu[n]]; Array[a, 105] (* Robert G. Wilson v, Apr 10 2012 *)
a[n_] := Sum[If[Mod[n, k] == 0, Re[Sqrt[MoebiusMu[k]]], 0], {k, 1, n}] (* Mats Granvik, Aug 10 2018 *)

a(n)=ceil((1<Charles R Greathouse IV, Nov 02 2011

a(n) = (1/2)*Sum_{ d divides n } abs(mu(d)) = 2^(A001221(n)-1) = A034444(n)/2, n > 1. - Vladeta Jovovic, Jan 25 2002
a(n) = phi(2^omega(n)) = A000010(2^A001221(n)). - Enrique Pérez Herrero, Apr 10 2012
Sum_{k=1..n} a(k) ~ 3*n*((log(n) + (2*gamma - 1))/ Pi^2 - 12*(zeta'(2)/Pi^4)), where gamma is the Euler-Mascheroni constant A001620. Equivalently, Sum_{k=1..n} a(k) ~ 3*n*(log(n) + 24*log(A) - 1 - 2*log(2*Pi)) / Pi^2, where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jan 30 2019
a(n) = Sum_{d|n} mu(d) * A018892(n/d). - Daniel Suteu, Jan 08 2021
Dirichlet g.f.: (zeta(s)^2/zeta(2*s) + 1)/2. - Amiram Eldar, Sep 09 2023

A260662 Decimal expansion of the generalized Glaisher-Kinkelin constant A(13).

Original entry on oeis.org

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

Offset: 1

G. C. Greubel, Nov 13 2015

Also known as the thirteenth Bendersky constant.

			1.2229442518081338726478999607277179885...

G. C. Greubel, Table of n, a(n) for n = 1..2001
Victor S. Adamchik, Polygamma functions of negative order, Journal of Computational and Applied Mathematics, Vol. 100, No. 2 (1998), pp. 191-199.
L. Bendersky, Sur la fonction gamma généralisée, Acta Mathematica , Vol. 61 (1933), pp. 263-322; alternative link.
Robert A. Van Gorder, Glaisher-type products over the primes, International Journal of Number Theory, Vol. 8, No. 2 (2012), pp. 543-550.
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant.

Cf. A019727 (A(0)), A074962 (A(1)), A243262 (A(2)), A243263 (A(3)), A243264 (A(4)), A243265 (A(5)), A266553 (A(6)), A266554 (A(7)), A266555 (A(8)), A266556 (A(9)), A266557 (A(10)), A266558 (A(11)), A266559 (A(12)), A260662 (A(13)), A266560 (A(14)), A266562 (A(15)), A266563 (A(16)), A266564 (A(17)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).

Cf. A001620, A260660, A260663.

N[Exp[(1/14)*HarmonicNumber[13]*BernoulliB[14] - Zeta'[-13]], 100]
Exp[N[(BernoulliB[14]/14)*(EulerGamma + Log[2*Pi] - Zeta'[14]/Zeta[14]), 200]]

A(k) = exp(H(k)*B(k+1)/(k+1) - zeta'(-k)), where B(k) is the k-th Bernoulli number, H(k) the k-th Harmonic number, and zeta'(x) is the derivative of the Riemann zeta function.
A(13) = exp((1/14)*HarmonicNumber(13)*Bernoulli(14) - RiemannZeta'(-13)).
A(13) = exp((B(14)/14)*(EulerGamma + Log(2*Pi) - (zeta'(14)/zeta(14)))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^14-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(14)/14 = 1/12 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024

A266554 Decimal expansion of the generalized Glaisher-Kinkelin constant A(7).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 31 2015

Also known as the 7th Bendersky constant.

			0.9899756533334170941753964830588692002082471514307453051285538624....

G. C. Greubel, Table of n, a(n) for n = 0..2000
Victor S. Adamchik, Polygamma functions of negative order, Journal of Computational and Applied Mathematics, Vol. 100, No. 2 (1998), pp. 191-199.
L. Bendersky, Sur la fonction gamma généralisée, Acta Mathematica , Vol. 61 (1933), pp. 263-322; alternative link.
Robert A. Van Gorder, Glaisher-type products over the primes, International Journal of Number Theory, Vol. 8, No. 2 (2012), pp. 543-550.
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant.

Cf. A019727 (A(0)), A074962 (A(1)), A243262 (A(2)), A243263 (A(3)), A243264 (A(4)), A243265 (A(5)), A266553 (A(6)), A266555 (A(8)), A266556 (A(9)), A266557 (A(10)), A266558 (A(11)), A266559 (A(12)), A260662 (A(13)), A266560 (A(14)), A266562 (A(15)), A266563 (A(16)), A266564 (A(17)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).

Cf. A001620, A013666, A259072, A027641, A027642, A001008, A002805.

Exp[N[(BernoulliB[8]/8)*(EulerGamma + Log[2*Pi] - Zeta'[8]/Zeta[8]), 200]]

A(k) = exp(H(k)*B(k+1)/(k+1) - zeta'(-k)), where B(k) is the k-th Bernoulli number, H(k) the k-th harmonic number, and zeta'(x) is the derivative of the Riemann zeta function.
A(7) = exp(H(7)*B(8)/8 - zeta'(-7)) = exp((B(8)/8)*(EulerGamma + log(2*Pi) - (zeta'(8)/zeta(8)))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^8-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(8)/8 = -1/240 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024

A266556 Decimal expansion of the generalized Glaisher-Kinkelin constant A(9).

Original entry on oeis.org

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

Offset: 1

G. C. Greubel, Dec 31 2015

Also known as the 9th Bendersky constant.

			1.018469929920992912170659049376672172308610190564074920380...

G. C. Greubel, Table of n, a(n) for n = 1..2002
Victor S. Adamchik, Polygamma functions of negative order, Journal of Computational and Applied Mathematics, Vol. 100, No. 2 (1998), pp. 191-199.
L. Bendersky, Sur la fonction gamma généralisée, Acta Mathematica , Vol. 61 (1933), pp. 263-322; alternative link.
Robert A. Van Gorder, Glaisher-type products over the primes, International Journal of Number Theory, Vol. 8, No. 2 (2012), pp. 543-550.
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant.

Cf. A019727 (A(0)), A074962 (A(1)), A243262 (A(2)), A243263 (A(3)), A243264 (A(4)), A243265 (A(5)), A266553 (A(6)), A266554 (A(7)), A266555 (A(8)), A266556 (A(9)), A266557 (A(10)), A266558 (A(11)), A266559 (A(12)), A260662 (A(13)), A266560 (A(14)), A266562 (A(15)), A266563 (A(16)), A266564 (A(17)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).

Cf. A001620, A013668, A266260, A027641, A027642, A001008, A002805.

Exp[N[(BernoulliB[10]/10)*(EulerGamma + Log[2*Pi] - Zeta'[10]/Zeta[10]), 200]]

A(k) = exp(H(k)*B(k+1)/(k+1) - zeta'(-k)), where B(k) is the k-th Bernoulli number, H(k) the k-th harmonic number, and zeta'(x) is the derivative of the Riemann zeta function.
A(9) = exp(H(9)*B(10)/10 - zeta'(-9)) = exp((B(10)/10)*(EulerGamma + log(2*Pi) - (zeta'(10)/zeta(10)))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^10-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(10)/10 = 1/132 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024

A266558 Decimal expansion of the generalized Glaisher-Kinkelin constant A(11).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 31 2015

Also known as the 11th Bendersky constant.

			0.950331248453288866514233841015331271597566403456173040861088881...

G. C. Greubel, Table of n, a(n) for n = 0..2000
Victor S. Adamchik, Polygamma functions of negative order, Journal of Computational and Applied Mathematics, Vol. 100, No. 2 (1998), pp. 191-199.
L. Bendersky, Sur la fonction gamma généralisée, Acta Mathematica , Vol. 61 (1933), pp. 263-322; alternative link.
Robert A. Van Gorder, Glaisher-type products over the primes, International Journal of Number Theory, Vol. 8, No. 2 (2012), pp. 543-550.
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant.

Cf. A019727 (A(0)), A074962 (A(1)), A243262 (A(2)), A243263 (A(3)), A243264 (A(4)), A243265 (A(5)), A266553 (A(6)), A266554 (A(7)), A266555 (A(8)), A266556 (A(9)), A266557 (A(10)), A266559 (A(12)), A260662 (A(13)), A266560 (A(14)), A266562 (A(15)), A266563 (A(16)), A266564 (A(17)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).

Cf. A001620, A013670, A266262, A027641, A027642, A001008, A002805.

Exp[N[(BernoulliB[12]/12)*(EulerGamma + Log[2*Pi] - Zeta'[12]/Zeta[12]), 200]]

A(k) = exp(H(k)*B(k+1)/(k+1) - zeta'(-k)), where B(k) is the k-th Bernoulli number, H(k) the k-th harmonic number, and zeta'(x) is the derivative of the Riemann zeta function.
A(11) = exp(H(11)*B(12)/12 - zeta'(-11)) = exp((B(12)/12)*(EulerGamma + log(2*Pi) - (zeta'(12)/zeta(12)))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^12-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(12)/12 = -691/32760 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024

A266562 Decimal expansion of the generalized Glaisher-Kinkelin constant A(15).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 31 2015

Also known as the 15th Bendersky constant.

			0.342830806132816736571711146340672378141726945483236877251076164....

G. C. Greubel, Table of n, a(n) for n = 0..2000
Victor S. Adamchik, Polygamma functions of negative order, Journal of Computational and Applied Mathematics, Vol. 100, No. 2 (1998), pp. 191-199.
L. Bendersky, Sur la fonction gamma généralisée, Acta Mathematica , Vol. 61 (1933), pp. 263-322; alternative link.
Robert A. Van Gorder, Glaisher-type products over the primes, International Journal of Number Theory, Vol. 8, No. 2 (2012), pp. 543-550.
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant.

Cf. A019727 (A(0)), A074962 (A(1)), A243262 (A(2)), A243263 (A(3)), A243264 (A(4)), A243265 (A(5)), A266553 (A(6)), A266554 (A(7)), A266555 (A(8)), A266556 (A(9)), A266557 (A(10)), A266558 (A(11)), A266559 (A(12)), A260662 (A(13)), A266560 (A(14)), A266563 (A(16)), A266564 (A(17)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).

Cf. A001620, A013674, A266270, A027641, A027642, A001008, A002805.

Exp[N[(BernoulliB[16]/16)*(EulerGamma + Log[2*Pi] - Zeta'[16]/Zeta[16]), 200]]

A(k) = exp(H(k)*B(k+1)/(k+1) - zeta'(-k)), where B(k) is the k-th Bernoulli number, H(k) the k-th harmonic number, and zeta'(x) is the derivative of the Riemann zeta function.
A(15) = exp(H(15)*B(16)/16 - zeta'(-15)) = exp((B(16)/16)*(EulerGamma + log(2*Pi) - zeta'(16)/zeta(16))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^16-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(16)/16 = -3617/8160 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024

A266564 Decimal expansion of the generalized Glaisher-Kinkelin constant A(17).

Original entry on oeis.org

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

Offset: 4

G. C. Greubel, Dec 31 2015

Also known as the 17th Bendersky constant.

			1596.53508575803855385145523662044194533166110061350444341....

G. C. Greubel, Table of n, a(n) for n = 4..2003
Victor S. Adamchik, Polygamma functions of negative order, Journal of Computational and Applied Mathematics, Vol. 100, No. 2 (1998), pp. 191-199.
L. Bendersky, Sur la fonction gamma généralisée, Acta Mathematica , Vol. 61 (1933), pp. 263-322; alternative link.
Robert A. Van Gorder, Glaisher-type products over the primes, International Journal of Number Theory, Vol. 8, No. 2 (2012), pp. 543-550.
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant.

Cf. A019727 (A(0)), A074962 (A(1)), A243262 (A(2)), A243263 (A(3)), A243264 (A(4)), A243265 (A(5)), A266553 (A(6)), A266554 (A(7)), A266555 (A(8)), A266556 (A(9)), A266557 (A(10)), A266558 (A(11)), A266559 (A(12)), A260662 (A(13)), A266560 (A(14)), A266562 (A(15)), A266563 (A(16)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).

Cf. A001620, A013676, A266272, A027641, A027642, A001008, A002805.

Exp[N[(BernoulliB[18]/18)*(EulerGamma + Log[2*Pi] - Zeta'[18]/Zeta[18]), 200]]

A(k) = exp(H(k)*B(k+1)/(k+1) - zeta'(-k)), where B(k) is the k-th Bernoulli number, H(k) the k-th harmonic number, and zeta'(x) is the derivative of the Riemann zeta function.
A(17) = exp(H(17)*B(18)/18 - zeta'(-17)) = exp((B(18)/18)*(EulerGamma + log(2*Pi) - zeta'(18)/zeta(18))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^18-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(18)/18 = 43867/14364 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024

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