A084448 -id:A084448 - cp's OEIS frontend

A266271 Decimal expansion of zeta'(-16) (the derivative of Riemann's zeta function at -16).

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

Offset: 1

G. C. Greubel, Dec 25 2015

			1.7730256608990963962477873441892944813554198276469991771639173077.....

G. C. Greubel, Table of n, a(n) for n = 1..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)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Mathematica
```
RealDigits[N[Zeta'[-16], 100]]
```

zeta'(-16) = (638512875*zeta(17))/(4*Pi^16) = - log(A(16)).

Equals (3617/2040)*(zeta(17)/zeta(16)).

Offset corrected by Rick L. Shepherd, May 21 2016

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

Original entry on oeis.org

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.

A258349 Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k-1)/2).

Original entry on oeis.org

1, 0, 1, 3, 7, 13, 28, 52, 107, 203, 396, 741, 1409, 2596, 4813, 8777, 15972, 28737, 51553, 91644, 162288, 285377, 499653, 869758, 1507615, 2599974, 4465606, 7635607, 13005252, 22061424, 37287395, 62788012, 105365891, 176211393, 293741195, 488101711, 808604106

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000294, A023871, A027999, A258347, A258348, A258350, A258351, A258352.

nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k*(k-1)/2),{k,1,nmax}],{x,0,nmax}],x]

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: binomial(n,2))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

a(n) ~ 1 / (2^(155/96) * 15^(11/96) * Pi^(1/24) * n^(59/96)) * exp(-Zeta'(-1)/2 - Zeta(3) / (8*Pi^2) - 75*Zeta(3)^3 / (2*Pi^8) - 15^(5/4) * Zeta(3)^2 / (2^(7/4) * Pi^5) * n^(1/4) - sqrt(15/2) * Zeta(3) / Pi^2 * sqrt(n) + 2^(7/4)*Pi / (3*15^(1/4)) * n^(3/4)), where Zeta(3) = A002117, Zeta'(-1) = A084448 = 1/12 - log(A074962).

G.f.: exp(Sum_{k>=1} (sigma_3(k) - sigma_2(k))*x^k/(2*k)). - Ilya Gutkovskiy, Aug 22 2018

A258352 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k-1)(k-2)/6).

Original entry on oeis.org

1, 0, 0, 1, 4, 10, 21, 39, 76, 145, 294, 581, 1169, 2276, 4435, 8494, 16237, 30768, 58221, 109466, 205223, 382658, 710808, 1314091, 2420437, 4439753, 8115645, 14781062, 26833241, 48550863, 87575527, 157480827, 282362462, 504819198, 900058558, 1600424247

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000335, A023872, A258346, A258347, A258348, A258349, A258350, A258351.

nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k*(k-1)*(k-2)/6),{k,1,nmax}],{x,0,nmax}],x]

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: binomial(n, 3))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

a(n) ~ Zeta(5)^(379/3600) / (2^(521/1800) * sqrt(5*Pi) * n^(2179/3600)) * exp(Zeta'(-1)/3 + Zeta(3)/(8*Pi^2) - Pi^16 / (3110400000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (216000 * Zeta(5)^2) - Zeta(3)^2/(90*Zeta(5)) + Zeta'(-3)/6 + (-Pi^12 / (10800000 * 2^(2/5) * Zeta(5)^(11/5)) + Pi^4 * Zeta(3) / (900 * 2^(2/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (36000 * 2^(4/5) * Zeta(5)^(7/5)) + Zeta(3) / (3 * 2^(4/5) * Zeta(5)^(2/5))) * n^(2/5) - Pi^4 / (180 * 2^(1/5) * Zeta(5)^(3/5)) * n^(3/5) + 5 * Zeta(5)^(1/5) / 2^(8/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663, Zeta'(-1) = A084448 = 1/12 - log(A074962), Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4.

A258347 Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k+1)).

Original entry on oeis.org

1, 2, 9, 28, 88, 250, 708, 1894, 4988, 12718, 31839, 77952, 187771, 444526, 1037522, 2387670, 5426996, 12188774, 27079379, 59541078, 129663636, 279801102, 598620511, 1270300142, 2674874760, 5591124784, 11605082733, 23926811840, 49016020317, 99798382290

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000294, A023871, A258341, A258348, A258349, A258350, A258351, A258352.

nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k*(k+1)),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ Pi^(1/12) / (2^(3/2) * 15^(7/48) * n^(31/48)) * exp(Zeta'(-1) - Zeta(3) / (4*Pi^2) + 75*Zeta(3)^3 / Pi^8 - 15^(5/4) * Zeta(3)^2 / (2*Pi^5) * n^(1/4) + sqrt(15) * Zeta(3) / Pi^2 * sqrt(n) + 4*Pi / (3*15^(1/4)) * n^(3/4)), where Zeta(3) = A002117, Zeta'(-1) = A084448 = 1/12 - log(A074962).

G.f.: exp(Sum_{k>=1} (sigma_2(k) + sigma_3(k))*x^k/k). - Ilya Gutkovskiy, Aug 22 2018

A258348 Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k-1)).

Original entry on oeis.org

1, 0, 2, 6, 15, 32, 79, 172, 397, 860, 1879, 3986, 8462, 17586, 36408, 74366, 150875, 303006, 604511, 1195872, 2350614, 4587484, 8898857, 17154278, 32883109, 62679852, 118858190, 224238730, 421021209, 786793776, 1463796383, 2711552690, 5002097398, 9190449808

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000294, A023871, A258344, A258347, A258349, A258350, A258351, A258352.

nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k*(k-1)),{k,1,nmax}],{x,0,nmax}],x]
Clear[a]; a[n_]:= a[n] = 1/n*Sum[(DivisorSigma[3, k]-DivisorSigma[2, k])*a[n-k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 100}] (* Vaclav Kotesovec, Apr 11 2016, following a suggestion of George Beck *)

a(n) ~ 1 / (2^(3/2) * 15^(5/48) * Pi^(1/12) * n^(29/48)) * exp(-Zeta'(-1) - Zeta(3)/(4*Pi^2) - 75*Zeta(3)^3 / Pi^8 - 15^(5/4) * Zeta(3)^2 / (2*Pi^5) * n^(1/4) - sqrt(15) * Zeta(3) / Pi^2 * sqrt(n) + 4*Pi / (3*15^(1/4)) * n^(3/4)), where Zeta(3) = A002117, Zeta'(-1) = A084448 = 1/12 - log(A074962).

G.f.: exp(Sum_{k>=1} (sigma_3(k) - sigma_2(k))*x^k/k). - Ilya Gutkovskiy, Aug 22 2018

A258350 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k+1)(k+2)).

Original entry on oeis.org

1, 6, 45, 260, 1410, 7026, 33212, 149190, 643959, 2681020, 10820736, 42468828, 162566956, 608302638, 2229485529, 8016901068, 28324233846, 98447346282, 336996263702, 1137220855428, 3786525025002, 12449461237388, 40446207528429, 129926295916884, 412912082761651

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A023872, A000335, A258342, A258347, A258348, A258349, A258351, A258352.

nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k*(k+1)*(k+2)),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ (3*Zeta(5))^(79/600) / (2^(21/200) * sqrt(5*Pi) * n^(379/600)) * exp(2*Zeta'(-1) - 3*Zeta(3)/(4*Pi^2) - Pi^16 / (518400000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (36000 * Zeta(5)^2) - Zeta(3)^2 / (15*Zeta(5)) + Zeta'(-3) + (Pi^12 / (1800000 * 2^(3/5) * 3^(1/5) * Zeta(5)^(11/5)) - Pi^4 * Zeta(3) / (150 * 2^(3/5) * 3^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (12000 * 2^(1/5) * 3^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(1/5) * (3*Zeta(5))^(2/5))) * n^(2/5) + Pi^4 / (30 * 2^(4/5) * (3*Zeta(5))^(3/5)) * n^(3/5) + 5 * (3*Zeta(5))^(1/5) / 2^(7/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663, Zeta'(-1) = A084448 = 1/12 - log(A074962), Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4.

cp's OEIS Frontend

A266271 Decimal expansion of zeta'(-16) (the derivative of Riemann's zeta function at -16).

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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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A258349 Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k-1)/2).

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A258352 Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k-1)*(k-2)/6).

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Mathematica

SageMath

Formula

A258347 Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k+1)).

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Author

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A258352 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k-1)(k-2)/6).

A258350 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k+1)(k+2)).