A248882 -id:A248882 - cp's OEIS frontend

A343323 Dirichlet g.f.: Product_{k>=2} (1 + k^(-s))^(k^3).

1, 8, 27, 92, 125, 432, 343, 1080, 1080, 2000, 1331, 5940, 2197, 5488, 6750, 12070, 4913, 20304, 6859, 27500, 18522, 21296, 12167, 76680, 23375, 35152, 42291, 75460, 24389, 135000, 29791, 132408, 71874, 78608, 85750, 309204, 50653, 109744, 118638, 355000, 68921, 370440, 79507, 292820

Offset: 1

Ilya Gutkovskiy, Apr 11 2021

nonn

Cf. A000578, A045778, A050368, A248882, A343283, A343322, A343324, A343325.

A304459 Coefficient of x^n in Product_{k>=1} (1+x^k)^(n^3).

Original entry on oeis.org

1, 1, 36, 3681, 770576, 276218900, 151479085752, 117975860569973, 123825991870849088, 168480096257782525419, 288418999876101261408100, 606652152400218992684850772, 1537897976017806908644807294656, 4624364862288125600795358272563097

Offset: 0

Vaclav Kotesovec, May 13 2018

nonn

In general, for m>=3, coefficient of x^n in Product_{k>=1} (1+x^k)^(n^m) is asymptotic to n^(m*n)/n!.

Cf. A248882, A270913, A304445, A304460.

nmax = 20; Table[SeriesCoefficient[Product[(1+x^k)^(n^3), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 20; Table[SeriesCoefficient[(QPochhammer[-1, x]/2)^(n^3), {x, 0, n}], {n, 0, nmax}]

a(n) ~ exp(n) * n^(2*n - 1/2) / sqrt(2*Pi).

A281157 Expansion of Product_{k>=1} (1 + x^k)^(k(2k^2+1)/3).

Original entry on oeis.org

1, 1, 6, 25, 78, 258, 800, 2463, 7344, 21511, 61677, 173980, 483319, 1323470, 3577605, 9553658, 25227727, 65918419, 170552866, 437196640, 1110945961, 2799689792, 7000246591, 17372882671, 42809388080, 104774554942, 254771953179, 615667051237, 1478934870484, 3532347875968

Offset: 0

Ilya Gutkovskiy, Jan 16 2017

nonn

Weigh transform of octahedral numbers (A005900).

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
Eric Weisstein's World of Mathematics, Octahedral Number

Cf. A005900, A248882, A258343.

nmax = 29; CoefficientList[Series[Product[(1 + x^k)^(k (2 k^2 + 1)/3), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^k)^(k*(2*k^2+1)/3).

a(n) ~ exp(-Zeta(3)^2 / (600*Zeta(5)) + (Zeta(3) / (4*(15*Zeta(5))^(2/5))) * n^(2/5) + (5*(15*Zeta(5))^(1/5) / 4) * n^(4/5)) * (3*Zeta(5))^(1/10) / (sqrt(Pi) * 2^(47/90) * 5^(2/5) * n^(3/5)). - Vaclav Kotesovec, Nov 09 2017

cp's OEIS Frontend

A343323 Dirichlet g.f.: Product_{k>=2} (1 + k^(-s))^(k^3).

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A304459 Coefficient of x^n in Product_{k>=1} (1+x^k)^(n^3).

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A281157 Expansion of Product_{k>=1} (1 + x^k)^(k(2k^2+1)/3).

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cp's OEIS Frontend

A343323 Dirichlet g.f.: Product_{k>=2} (1 + k^(-s))^(k^3).

Views

Author

Keywords

Crossrefs

A304459 Coefficient of x^n in Product_{k>=1} (1+x^k)^(n^3).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A281157 Expansion of Product_{k>=1} (1 + x^k)^(k*(2*k^2+1)/3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A281157 Expansion of Product_{k>=1} (1 + x^k)^(k(2k^2+1)/3).