A258352 -id:A258352 - cp's OEIS frontend

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

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

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

A258346 Expansion of Product_{k>=1} (1+x^k)^(k(k-1)(k-2)/6).

Original entry on oeis.org

1, 0, 0, 1, 4, 10, 20, 39, 72, 144, 280, 567, 1112, 2187, 4204, 8073, 15309, 28986, 54548, 102286, 190881, 354717, 656194, 1208712, 2217624, 4052633, 7379630, 13390098, 24215587, 43649482, 78435884, 140513905, 250988186, 447037367, 794031641, 1406585604

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

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

Cf. A248882, A028377, A258341, A258342, A258343, A258344, A258345, A258352.

nmax=50; CoefficientList[Series[Product[(1+x^k)^(k*(k-1)*(k-2)/6),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ (3*Zeta(5))^(1/10) / (2^(523/720) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-2401 * Pi^16 / (10497600000000 * Zeta(5)^3) + 49*Pi^8 * Zeta(3) / (16200000 * Zeta(5)^2) - Zeta(3)^2 / (150*Zeta(5)) + (-343*Pi^12 / (2430000000 * 2^(3/5) * 15^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (4500 * 2^(3/5) * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (1080000 * 2^(1/5) * 15^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(6/5) * (15*Zeta(5))^(2/5))) * n^(2/5) - 7*Pi^4 / (180 * 2^(4/5) * (15*Zeta(5))^(3/5)) * n^(3/5) + 5*(15*Zeta(5))^(1/5) / 2^(12/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663.

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.

A258351 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k-1)(k-2)).

Original entry on oeis.org

1, 0, 0, 6, 24, 60, 141, 354, 996, 2720, 7194, 18306, 46154, 115506, 288195, 713210, 1749732, 4253148, 10259302, 24573390, 58491312, 138371354, 325415727, 760899396, 1769420183, 4093054602, 9420739965, 21578842582, 49199229066, 111672215658, 252381169048

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

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

Cf. A000335, A023872, A258345, A258347, A258348, A258349, A258350, 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.

A264924 G.f.: 1 / Product_{n>=0} (1 - x^(n+4))^((n+1)(n+2)(n+3)/3!).

Original entry on oeis.org

1, 0, 0, 0, 1, 4, 10, 20, 36, 60, 104, 180, 336, 620, 1174, 2160, 3961, 7100, 12690, 22424, 39651, 69820, 122970, 215904, 378470, 660872, 1150740, 1996200, 3452685, 5952916, 10237576, 17559460, 30049285, 51301020, 87390872, 148534232, 251916041, 426329040, 720003646, 1213481344, 2041155052, 3426721080

Offset: 0

Paul D. Hanna, Nov 28 2015

nonn

Number of partitions of n objects of 4 colors, where each part must contain at least one of each color. [Conjecture - see comment by Franklin T. Adams-Watters in A052847].

			G.f.: A(x) = 1 + x^4 + 4*x^5 + 10*x^6 + 20*x^7 + 36*x^8 + 60*x^9 + 104*x^10 + 180*x^11 +...
where
1/A(x) = (1-x^4) * (1-x^5)^4 * (1-x^6)^10 * (1-x^7)^20 * (1-x^8)^35 * (1-x^9)^56 * (1-x^10)^84 * (1-x^11)^120 * (1-x^12)^165 *...
Also,
log(A(x)) = (x/(1-x))^4 + (x^2/(1-x^2))^4/2 + (x^3/(1-x^3))^4/3 + (x^4/(1-x^4))^4/4 + (x^5/(1-x^5))^4/5 + (x^6/(1-x^6))^4/6 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A052847, A264923, A264925, A264926.

Cf. A000335, A255050, A258352.

nmax = 50; CoefficientList[Series[Product[1/(1-x^k)^((k-3)*(k-2)*(k-1)/6), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Dec 09 2015 *)

{a(n) = my(A=1); A = prod(k=0,n, 1/(1 - x^(k+4) +x*O(x^n) )^((k+1)*(k+2)*(k+3)/3!) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

{a(n) = my(A=1); A = exp( sum(k=1,n+1, (x^k/(1 - x^k))^4 /k +x*O(x^n) ) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

{L(n) = sumdiv(n,d, d*(d-1)*(d-2)*(d-3)/3! )}
{a(n) = my(A=1); A = exp( sum(k=1,n+1, L(k) * x^k/k +x*O(x^n) ) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} ( x^n/(1-x^n) )^4 /n ).
G.f.: exp( Sum_{n>=1} L(n) * x^n/n ), where L(n) = Sum_{d|n} d*(d-1)*(d-2)*(d-3)/3!.
a(n) ~ Zeta(5)^(109/3600) / (2^(791/1800) * n^(1909/3600) * sqrt(5*Pi)) * exp(11*Zeta'(-1)/6 + log(2*Pi)/2 + Zeta(3)/(4*Pi^2) - Pi^16/(194400000 * Zeta(5)^3) + 11*Pi^8 * Zeta(3)/(108000 * Zeta(5)^2) - Pi^6/(1800*Zeta(5)) - 121*Zeta(3)^2/(360*Zeta(5)) + Zeta'(-3)/6 + (-Pi^12/(1350000 * 2^(2/5) * Zeta(5)^(11/5)) + 11*Pi^4 * Zeta(3)/(900 * 2^(2/5) * Zeta(5)^(6/5)) - Pi^2/(3*2^(7/5) * Zeta(5)^(1/5))) * n^(1/5) + (-Pi^8/(9000 * 2^(4/5) * Zeta(5)^(7/5)) + 11*Zeta(3)/(3*2^(9/5) * Zeta(5)^(2/5))) * n^(2/5) - Pi^4/(90 * 2^(1/5) * Zeta(5)^(3/5)) * n^(3/5) + 5*Zeta(5)^(1/5) / 2^(8/5) * n^(4/5)). - Vaclav Kotesovec, Dec 09 2015

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

SageMath

Formula

A258346 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)*(k-2)/6).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A258351 Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k-1)*(k-2)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A264924 G.f.: 1 / Product_{n>=0} (1 - x^(n+4))^((n+1)*(n+2)*(n+3)/3!).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

A258346 Expansion of Product_{k>=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)).

A258351 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k-1)(k-2)).

A264924 G.f.: 1 / Product_{n>=0} (1 - x^(n+4))^((n+1)(n+2)(n+3)/3!).