A259068 -id:A259068 - cp's OEIS frontend

A266273 Decimal expansion of zeta'(-18) (the derivative of Riemann's zeta function at -18) (negated).

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

Offset: 2

G. C. Greubel, Dec 25 2015

			-13.74276825021405443522056419051855107309537215770498560....

G. C. Greubel, Table of n, a(n) for n = 2..1500

Cf. A075700 (zeta'(0)), A084448 (zeta'(-1)), A240966 (zeta'(-2)), A259068 (zeta'(-3)), A259069 (zeta'(-4)), A259070 (zeta'(-5)), A259071 (zeta'(-6)), A259072 (zeta'(-7)), A259073 (zeta'(-8)), A266260 (zeta'(-9)), A266261 (zeta'(-10)), A266262 (zeta'(-11)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Mathematica
```
RealDigits[N[Zeta'[-18], 100]]
```

zeta'(-18) = -(97692469875*zeta(19))/(8*Pi^18) = - log(A(18)).

Equals -(43867/3192)*(zeta(19)/zeta(18)).

Offset corrected by Rick L. Shepherd, May 30 2016

A266274 Decimal expansion of zeta'(-19) (the derivative of Riemann's zeta function at -19) (negated).

Original entry on oeis.org

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

Offset: 2

G. C. Greubel, Dec 26 2015

			-29.965529831392351939431865297274201791908226109115565915881....

G. C. Greubel, Table of n, a(n) for n = 2..1500

Cf. A075700 (zeta'(0)), A084448 (zeta'(-1)), A240966 (zeta'(-2)), A259068 (zeta'(-3)), A259069 (zeta'(-4)), A259070 (zeta'(-5)), A259071 (zeta'(-6)), A259072 (zeta'(-7)), A259073 (zeta'(-8)), A266260 (zeta'(-9)), A266261 (zeta'(-10)), A266262 (zeta'(-11)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266275 (zeta'(-20)).

Mathematica
```
RealDigits[N[Zeta'[-19], 100]]
```

zeta'(-n) = (BernoulliB(n+1)*HarmonicNumber(n))/(n+1) - log(A(n)), where A(n) is the n-th Bendersky constant.

zeta'(-19) = -48069674759189/512143632000 - log(A(19)).

Offset corrected by Rick L. Shepherd, May 30 2016

A266275 Decimal expansion of zeta'(-20) (the derivative of Riemann's zeta function at -20).

Original entry on oeis.org

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

Offset: 3

G. C. Greubel, Dec 26 2015

			132.28099750421251452709821158578551868064800999955031458847450192414...

G. C. Greubel, Table of n, a(n) for n = 3..1500

Cf. A075700 (zeta'(0)), A084448 (zeta'(-1)), A240966 (zeta'(-2)), A259068 (zeta'(-3)), A259069 (zeta'(-4)), A259070 (zeta'(-5)), A259071 (zeta'(-6)), A259072 (zeta'(-7)), A259073 (zeta'(-8)), A266260 (zeta'(-9)), A266261 (zeta'(-10)), A266262 (zeta'(-11)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)).

Mathematica
```
RealDigits[N[Zeta'[-20], 100]]
```

zeta'(-20) = (9280784638125*zeta(21))/(8*Pi^20) = - log(A(20)).

Equals (174611/1320)*(zeta(21)/zeta(20)).

Offset corrected by Rick L. Shepherd, May 30 2016

A261506 Decimal expansion of -zeta'(4).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Aug 22 2015

			0.06891126589612537984882936558744082715001637487137...

J. B. Rosser, L. Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. J. Math. 6 (1) (1962) 64-94, Table IV

Cf. A075700 (0), A073002 (2), A244115 (3).

Cf. A084448 (-1), A240966 (-2), A259068 (-3), A259069 (-4), A259070 (-5), A259071 (-6), A259072 (-7), A259073 (-8).

Flatten[{0, RealDigits[-Zeta'[4], 10, 105][[1]]}]

Sum_{n>=1} log(n) / n^4.

A206623 G.f.: Product_{n>0} ( (1+x^n)/(1-x^n) )^(n^3).

Original entry on oeis.org

1, 2, 18, 88, 398, 1768, 7508, 30644, 121310, 467234, 1756080, 6457168, 23274788, 82381584, 286760344, 982874120, 3320800590, 11070619228, 36446345198, 118581503192, 381552358872, 1214868568728, 3829841265428, 11959828895612, 37013411304892, 113570015855642

Offset: 0

Paul D. Hanna, Feb 12 2012

nonn

Convolution of A023872 and A248882. - Vaclav Kotesovec, Aug 19 2015

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 88*x^3 + 398*x^4 + 1768*x^5 + 7508*x^6 +...
where A(x) = (1+x)/(1-x) * (1+x^2)^8/(1-x^2)^8 * (1+x^3)^27/(1-x^3)^27 *...
Also, A(x) = Euler transform of [2,15,54,120,250,405,686,960,1458,...]:
A(x) = 1/((1-x)^2*(1-x^2)^15*(1-x^3)^54*(1-x^4)^120*(1-x^5)^250*(1-x^6)^405*...).

Seiichi Manyama, Table of n, a(n) for n = 0..4595 (terms 0..1000 from Vaclav Kotesovec)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 23.

Cf. A156616, A206622, A206624, A001159 (sigma_4).

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(k^3), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2015 *)

{a(n)=polcoeff(prod(m=1,n+1,((1+x^m)/(1-x^m+x*O(x^n)))^(m^3)),n)}

{a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m, 4)-sigma(m, 4))/8*x^m/m)+x*O(x^n)), n)}

{a(n)=local(InvEulerGF=x*(2+15*x+46*x^2+60*x^3+46*x^4+15*x^5+2*x^6)/(1-x^2+x*O(x^n))^4);polcoeff(1/prod(k=1,n,(1-x^k+x*O(x^n))^polcoeff(InvEulerGF,k)),n)}
for(n=0,30,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} (sigma_4(2*n) - sigma_4(n))/8 * x^n/n ), where sigma_4(n) is the sum of 4th powers of divisors of n (A001159).
Inverse Euler transform has g.f.: x*(2 + 15*x + 46*x^2 + 60*x^3 + 46*x^4 + 15*x^5 + 2*x^6)/(1-x^2)^4.
a(n) ~ (93*Zeta(5))^(59/600) * exp(5/4 * (93*Zeta(5)/2)^(1/5) * n^(4/5) + Zeta'(-3)) / (2^(59/100) * sqrt(5*Pi) * n^(359/600)), where Zeta(5) = A013663, Zeta'(-3) = A259068. - Vaclav Kotesovec, Aug 19 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.

A057527 4th level factorials: product of first n superduperfactorials.

Original entry on oeis.org

1, 1, 2, 48, 331776, 79254226206720, 471092427871945743012986880000, 351177419973413722592573060611594181593855426560000000000

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.

			a(4) =((4!*3!*2!*1!)*(3!*2!*1!)*(2!*1!)*(1!)) * ((3!*2!*1!)*(2!*1!)*(1!)) * ((2!*1!)*(1!)) * ((1!)) =24*6^3*2^6*1^10 =331776

Cf. A000142, A000178, A055462, A057528, A260404 for first, second, third, fifth and sixth level factorials.

Table[Product[i^Binomial[n-i+3,3],{i,1,n}],{n,0,10}] (* Vaclav Kotesovec, Jul 24 2015 *)
Nest[FoldList[Times,#]&,Range[0,8]!,3] (* Harvey P. Dale, Jan 08 2024 *)

a(n) =a(n-1)*A055462(n) =Product[i^A000332(n-i)] over 1<=i<=n.

a(n) ~ exp(11/72 - 5*n/6 - 4*n^2/3 - 11*n^3/18 - 25*n^4/288 + Zeta(3)*(n+2) / (8*Pi^2) + Zeta'(-3)/6) * n^(251/720 + n + 11*n^2/12 + n^3/3 + n^4/24) * (2*Pi)^((n+1)*(n+2)*(n+3)/12) / A^(11/6 + 2*n + n^2/2), where Zeta(3) = A002117, Zeta'(-3) = A259068 = 0.0053785763577743011444169742104138428956644397... and A = A074962 = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 24 2015

A255511 Decimal expansion of a constant related to A255358.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Feb 24 2015

			4.113740552015338123052453340090368136395763815194771589658140463089224...

Cf. A255358, A255504, A255438, A255439.

Cf. A074962, A243262, A243263, A243264, A243265.

Equals limit n->infinity (Product_{k=0..n} (k^3)!) / (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)).

Equals (2*Pi)^(3/4) * exp(-11/240 - 3*Zeta'(-3)) * Product_{n>=1} ((n^3)!/stirling(n^3)), where stirling(n^3) = sqrt(2*Pi) * n^(3*n^3 + 3/2) / exp(n^3) is the Stirling approximation of (n^3)! and Zeta'(-3) = A259068. - Vaclav Kotesovec, Apr 20 2016

A271170 Decimal expansion of the logarithm of the generalized Glaisher-Kinkelin constant A(3) (negated).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Apr 01 2016

The logarithm of the third Bendersky constant.

			-0.02065635413555207892219475198819162067344221752007...

G. C. Greubel, Table of n, a(n) for n = 0..2000

log(A(b)): A225746 (b=1), (-1) * A240966 (b=2).

Cf. A243263, A259068.

Join[{0}, RealDigits[(BernoulliB[4]/4)*(EulerGamma + Log[2*Pi] - Zeta'[4]/Zeta[4]), 10, 100] // First]

Equals (Bernoulli(4)/4)*(EulerGamma + log(2*Pi) - (Zeta'(4)/Zeta(4))).

log(A(3)) = HarmonicNumber(3)*Bernoulli(4)/4 - Zeta'(-3).

cp's OEIS Frontend

A266273 Decimal expansion of zeta'(-18) (the derivative of Riemann's zeta function at -18) (negated).

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A266274 Decimal expansion of zeta'(-19) (the derivative of Riemann's zeta function at -19) (negated).

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A266275 Decimal expansion of zeta'(-20) (the derivative of Riemann's zeta function at -20).

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A261506 Decimal expansion of -zeta'(4).

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A206623 G.f.: Product_{n>0} ( (1+x^n)/(1-x^n) )^(n^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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A057527 4th level factorials: product of first n superduperfactorials.

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