A258348 -id:A258348 - cp's OEIS frontend

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

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

A258344 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)).

Original entry on oeis.org

1, 0, 2, 6, 13, 32, 69, 160, 344, 760, 1601, 3384, 7022, 14434, 29361, 59140, 118089, 233754, 459293, 895382, 1733904, 3334914, 6374654, 12111632, 22881777, 42993244, 80362496, 149464404, 276657082, 509740278, 935046158, 1707916988, 3106810873, 5629121054

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

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

Cf. A027998, A028377, A027999, A258341, A258342, A258343, A258345, A258346, A258348.

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

a(n) ~ 7^(1/8) / (2^(43/24) * 15^(1/8) * n^(5/8)) * exp(-2025*Zeta(3)^3 / (49*Pi^8) - 135*(15/14)^(1/4) * Zeta(3)^2 / (14*Pi^5) * n^(1/4) - 3*sqrt(15/14) * Zeta(3) / Pi^2 * sqrt(n) + 2*(14/15)^(1/4)*Pi/3 * n^(3/4)), where Zeta(3) = A002117.

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

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.

A295828 Expansion of Product_{k>=1} 1/(1 - x^k)^(2k(2*k-1)).

Original entry on oeis.org

1, 2, 15, 58, 235, 862, 3122, 10664, 35639, 115164, 363806, 1122050, 3393316, 10068006, 29374056, 84347944, 238713339, 666419456, 1836986443, 5003473866, 13476019215, 35912177618, 94746481999, 247597696802, 641205816641, 1646268490598, 4192059724668, 10590937903412, 26556243826240

Offset: 0

Ilya Gutkovskiy, Nov 28 2017

nonn

Euler transform of A002939.

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Cf. A002939, A161870, A253289, A258347, A258348, A278767.

nmax = 28; CoefficientList[Series[Product[1/(1 - x^k)^(2 k (2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[2 d^2 (2 d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A002939(k).

a(n) ~ exp(2^(5/2) * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) - Zeta(3) * sqrt(15*n) / Pi^2 - 15^(5/4) * Zeta(3)^2 * n^(1/4) / (2^(3/2) * Pi^5) - Zeta(3) / Pi^2 - 75*Zeta(3)^3 / (2*Pi^8) - 1/6) * A^2 / (2^(4/3) * 15^(1/12) * Pi^(1/6) * n^(7/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 28 2017

cp's OEIS Frontend

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

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A258344 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)).

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

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

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

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

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A295828 Expansion of Product_{k>=1} 1/(1 - x^k)^(2*k*(2*k-1)).

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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)).

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

A295828 Expansion of Product_{k>=1} 1/(1 - x^k)^(2k(2*k-1)).