A075700 -id:A075700 - cp's OEIS frontend

A052847 G.f.: 1 / Product_{k>=1} (1-x^k)^(k-1).

1, 0, 1, 2, 4, 6, 12, 18, 33, 52, 88, 138, 229, 354, 568, 880, 1378, 2110, 3260, 4942, 7527, 11320, 17031, 25394, 37842, 55956, 82630, 121300, 177677, 258980, 376626, 545352, 787784, 1133764, 1627657, 2329020, 3324559, 4731396, 6717774, 9512060

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Euler transform of sequence [0,1,2,3,...]. - Michael Somos, Jul 02 2004
Number of partitions of n objects of 2 colors, where each part must contain at least one of each color. - Franklin T. Adams-Watters, Jan 23 2006
Number of partitions of n without 1s, one kind of 2s, two kinds of 3s, etc. - Joerg Arndt, Jul 31 2011
From Vaclav Kotesovec, Oct 17 2015: (Start)
In general, if v>=0 and g.f. = Product_{k>=1} 1/(1-x^(k+v))^k, then a(n) ~ d1(v) * n^(v^2/6 - 25/36) * exp(-Pi^4 * v^2 / (432*Zeta(3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2^(2/3) - v * Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3))) / (sqrt(3*Pi) * 2^(v^2/6 + 11/36) * Zeta(3)^(v^2/6 - 7/36)), where Zeta(3) = A002117.
d1(v) = exp(Integral_{x=0..infinity} (1/(x*exp((v-1)*x) * (exp(x)-1)^2) - (6*v^2-1) / (12*x*exp(x)) + v/x^2 - 1/x^3) dx).
d1(v) = (exp(Zeta'(-1) - v*Zeta'(0))) / Product_{j=0..v-1} j!, where Zeta'(0) = -A075700, Zeta'(-1) = A084448 and Product_{j=0..v-1} j! = A000178(v-1).
d1(v) = exp(1/12) * (2*Pi)^(v/2) / (A * G(v+1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.
(End)

			1 + x^2 + 2*x^3 + 4*x^4 + 6*x^5 + 12*x^6 + 18*x^7 + 33*x^8 + 52*x^9 + ...
From _Gus Wiseman_, Jan 22 2019: (Start)
The partitions described in Franklin T. Adams-Watters's comment are (n = 2 through 6):
  {{12}}  {{112}}  {{1112}}    {{11112}}    {{111112}}
          {{122}}  {{1122}}    {{11122}}    {{111122}}
                   {{1222}}    {{11222}}    {{111222}}
                   {{12}{12}}  {{12222}}    {{112222}}
                               {{12}{112}}  {{122222}}
                               {{12}{122}}  {{112}{112}}
                                            {{112}{122}}
                                            {{12}{1112}}
                                            {{12}{1122}}
                                            {{12}{1222}}
                                            {{122}{122}}
                                            {{12}{12}{12}}
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 815
Vaclav Kotesovec, Graph - The asymptotic ratio

Cf. A005380, A000203, A001157, A052812.
Cf. A000219 (v=0), A052847 (v=1), A263358 (v=2), A263359 (v=3), A263360 (v=4), A263361 (v=5), A263362 (v=6), A263363 (v=7), A263364 (v=8).
Cf. A054974, A321407, A321760, A323654, A323655, A323656.

spec := [S,{B=Sequence(Z,1 <= card),C=Prod(B,B),S= Set(C)},unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:=etr(n-> n-1): seq(a(n), n=0..50); # Vaclav Kotesovec, Mar 04 2015 after Alois P. Heinz

Clear[a]; a[n_]:= a[n] = 1/n*Sum[(DivisorSigma[2,k]-DivisorSigma[1,k])*a[n-k],{k,1,n}]; a[0]=1; Table[a[n],{n,0,100}] (* Vaclav Kotesovec, Mar 04 2015 *)
nmax = 40; CoefficientList[Series[Product[1/(1-x^(k+1))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 16 2015 *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, (1 - x^k + x*O(x^n))^(k-1)), n))}

a(n) = 1/n*Sum_{k=1..n} (sigma[2](k)-sigma[1](k))*a(n-k).
G.f.: exp( Sum_{k>0} ( x^k / (1 - x^k) )^2 / k ).
G.f.: exp( sum(n>=0, (sigma[2](n)-sigma[1](n)) *x^n/n ) ). - Joerg Arndt, Jul 31 2011
a(n) ~ 2^(1/36) * Zeta(3)^(1/36) * exp(1/12 - Pi^4/(432*Zeta(3)) - Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A * 3^(1/2) * n^(19/36)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 07 2015

Edited by Vladeta Jovovic, Sep 10 2002

A240966 Decimal expansion of zeta'(-2) (the derivative of Riemann's zeta function at -2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Aug 05 2014

			-0.030448457058393270780251530471154776647000483544973936252971889859...

Eric Weisstein's MathWorld, Riemann Zeta Function.

Cf. A084448 (zeta'(-1)), A075700 (zeta'(0)), A073002 (zeta'(2)), A244115 (zeta'(3)).

Join[{0}, RealDigits[-Zeta[3]/(4*Pi^2), 10, 103] // First]

zeta'(-2) = -zeta(3)/(4*Pi^2).

Equals -log(A243262). - Vaclav Kotesovec, Feb 22 2015

A266262 Decimal expansion of zeta'(-11) (the derivative of Riemann's zeta function at -11) (negated).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 25 2015

			-0.012752984479966656113522525488725798156238937049874292793246366661...

G. C. Greubel, Table of n, a(n) for n = 0..1500

Cf. A075700 (zeta'(0)), A084448 (zeta'(-1)), A240966 (zeta'(-2)), A259068 (zeta'(-3)), A259070 (zeta'(-5)), A259071 (zeta'(-6)), A259072 (zeta'(-7)), A259073 (zeta'(-8)), A266260 (zeta'(-9)), A266261 (zeta'(-10)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Join[{0}, RealDigits[Zeta'[-11], 10, 100] // First]

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

zeta'(-11) = - 57844301/908107200 - log(A(11)).

Keyword cons added by Rick L. Shepherd, May 29 2016

A260660 Decimal expansion of zeta'(-13) (the derivative of Riemann's zeta function at -13).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Nov 13 2015

			0.06374987374457688028603868147333505564882735...

G. C. Greubel, Table of n, a(n) for n = 0..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)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

N[Zeta'[-13]]
Join[{0}, RealDigits[Zeta'[-13], 10, 1500] // First]

PARI
```
zeta'(-13) \\ Altug Alkan, Nov 13 2015
```

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'(-13) = (1145993/4324320) - log(A(13)).
zeta'(-13) = 1145993/4324320 - gamma/12 - log(2*Pi)/12 + 6081075*Zeta'(14) / (8*Pi^14), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 05 2015

A266260 Decimal expansion of zeta'(-9) (the derivative of Riemann's zeta function at -9).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 25 2015

			0.0031301453197885727549257682907854467026693658654815.....

G. C. Greubel, Table of n, a(n) for n = 0..1501

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)), 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)), A266275 (zeta'(-20)).

Join[{0, 0}, RealDigits[Zeta'[-9], 10, 100] // First]
N[Zeta'[-9], 100]

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

zeta'(-9) = 7129/332640 - log(A(9)).

A266270 Decimal expansion of zeta'(-15) (the derivative of Riemann's zeta function at -15).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 25 2015

			-0.400319302807725593843580317520320367201261286266232944284106942....

G. C. Greubel, Table of n, a(n) for n = 0..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)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Mathematica
```
RealDigits[N[Zeta'[-15], 100]]
```

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

zeta'(-15) = -4325053069/2940537600 - log(A(15)).

A266272 Decimal expansion of zeta'(-17) (the derivative of Riemann's zeta function at -17).

Original entry on oeis.org

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

Offset: 1

G. C. Greubel, Dec 25 2015

			3.1286453321241578756844526391533305482263390775654797424916577061....

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)), A266271 (zeta'(-16)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Mathematica
```
RealDigits[N[Zeta'[-17], 100]]
```

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

zeta'(-17) = 1848652896341/175991175360 - log(A(17)).

Offset corrected by Rick L. Shepherd, May 21 2016

A266261 Decimal expansion of zeta'(-10) (the derivative of Riemann's zeta function at -10).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 25 2015

			-0.0189299263381403742289805022903467952319852580951695558

G. C. Greubel, Table of n, a(n) for n = 0..1500

Cf. A075700 (zeta'(0)), A084448 (zeta'(-1)), A240966 (zeta'(-2)), A259068 (zeta'(-3)), A259070 (zeta'(-5)), A259071 (zeta'(-6)), A259072 (zeta'(-7)), A259073 (zeta'(-8)), A266260 (zeta'(-9)), 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)), A266275 (zeta'(-20)).

Join[{0}, RealDigits[-(5/264)*(Zeta[11]/Zeta[10]), 10, 100] // First]

zeta'(-10) = -14175*zeta(11)/(8*Pi^10) = log(A(10)).

Equals -(5/264)*(zeta(11)/zeta(10)).

A266263 Decimal expansion of zeta'(-12) (the derivative of Riemann's zeta function at -12).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 25 2015

			0.06327058334146300059518230123430776751141818475323637667956594567...

G. C. Greubel, Table of n, a(n) for n = 0..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)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Join[{0}, RealDigits[(691/10920)*(Zeta[13]/Zeta[12]), 10, 100] // First]

zeta'(-12) = (-467775*Zeta(13))/(8*Pi^12) = - log(A(12)).

Equals (691/10920)*(zeta(13)/zeta(12)).

A266264 Decimal expansion of zeta'(-14) (the derivative of Riemann's zeta function at -14).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 25 2015

			-0.29165772474387352032122400307025066697026303853309083214990....

G. C. Greubel, Table of n, a(n) for n = 0..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)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Mathematica
```
RealDigits[N[Zeta'[-14], 100]]
```

zeta'(-14) = - (42567525*zeta(15))/(16*Pi^14) = - log(A(14)).

Equals -(7/24)*(zeta(15)/zeta(14)).

cp's OEIS Frontend

A052847 G.f.: 1 / Product_{k>=1} (1-x^k)^(k-1).

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A240966 Decimal expansion of zeta'(-2) (the derivative of Riemann's zeta function at -2).

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A266262 Decimal expansion of zeta'(-11) (the derivative of Riemann's zeta function at -11) (negated).

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A260660 Decimal expansion of zeta'(-13) (the derivative of Riemann's zeta function at -13).

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A266260 Decimal expansion of zeta'(-9) (the derivative of Riemann's zeta function at -9).

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A266270 Decimal expansion of zeta'(-15) (the derivative of Riemann's zeta function at -15).

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A266272 Decimal expansion of zeta'(-17) (the derivative of Riemann's zeta function at -17).

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A266261 Decimal expansion of zeta'(-10) (the derivative of Riemann's zeta function at -10).

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