A246584 -id:A246584 - cp's OEIS frontend

A246585 Zeroth trisection of A246584.

1, 12, 92, 470, 2020, 7456, 24962, 76736, 221156, 602200, 1565088, 3902344, 9389022, 21876192, 49542464, 109333520, 235717092, 497416856, 1029271544, 2091480776, 4179123296, 8220922592, 15937506968, 30477650240, 57540075230, 107327212780, 197923648864

Offset: 0

N. J. A. Sloane, Sep 03 2014

nonn

Michael D. Hirschhorn, A note on overcubic partitions, New Zealand J. Math., 42:229-234, 2012.
Bernard L. S. Lin, Arithmetic properties of overcubic partition pairs, Electronic Journal of Combinatorics 21(3) (2014), #P3.35.

Cf. A246584, A246586, A246587.

a(n) ~ exp(3*Pi*sqrt(n/2)) / (2^(19/4)*sqrt(3)*n^(5/4)). - Vaclav Kotesovec, Aug 16 2019

A246586 First trisection of A246584.

Original entry on oeis.org

2, 26, 160, 784, 3152, 11290, 36556, 110012, 310482, 832224, 2131072, 5249356, 12488368, 28816368, 64676460, 141598304, 303031202, 635197072, 1306210848, 2639071792, 5245173264, 10266881562, 19811351308, 37720810240, 70922550372, 131777935396, 242124952896

Offset: 0

N. J. A. Sloane, Sep 03 2014

nonn

Michael D. Hirschhorn, A note on overcubic partitions, New Zealand J. Math., 42:229-234, 2012.
Bernard L. S. Lin, Arithmetic properties of overcubic partition pairs, Electronic Journal of Combinatorics 21(3) (2014), #P3.35.

Cf. A246584, A246585, A246587.

a(n) ~ exp(3*Pi*sqrt(n/2)) / (2^(19/4)*sqrt(3)*n^(5/4)). - Vaclav Kotesovec, Aug 16 2019

A246587 Second trisection of A246584.

Original entry on oeis.org

6, 48, 282, 1260, 4896, 16836, 53232, 156384, 433776, 1143696, 2890266, 7032576, 16557084, 37838052, 84206724, 182913216, 388685430, 809399280, 1654446816, 3323927340, 6572070528, 12801615744, 24590359284, 46619988384, 87302773392, 161597518272, 295849759728

Offset: 0

N. J. A. Sloane, Sep 03 2014

nonn

Michael D. Hirschhorn, A note on overcubic partitions, New Zealand J. Math., 42:229-234, 2012.
Bernard L. S. Lin, Arithmetic properties of overcubic partition pairs, Electronic Journal of Combinatorics 21(3) (2014), #P3.35.

Cf. A246584, A246585, A246586.

a(n) ~ exp(3*Pi*sqrt(n/2)) / (2^(19/4)*sqrt(3)*n^(5/4)). - Vaclav Kotesovec, Aug 16 2019

a(n) = A246584(3*n+2). - Alois P. Heinz, Oct 27 2022

A327048 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) / ((1 - x^k) (1 - x^(2k)) (1 - x^(3*k))).

Original entry on oeis.org

1, 2, 6, 14, 30, 60, 120, 220, 402, 710, 1224, 2064, 3438, 5596, 9012, 14304, 22422, 34740, 53330, 80960, 121908, 181976, 269484, 396072, 578232, 838258, 1207896, 1730058, 2463900, 3490020, 4918572, 6897012, 9626610, 13375776, 18504852, 25494456, 34985530

Offset: 0

Vaclav Kotesovec, Aug 16 2019

nonn

Convolution of A327045 and A327042.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A015128, A246584, A327049, A327050.

Cf. A301554.

nmax = 50; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k)) * (1+x^(3*k)) / ((1-x^k) * (1-x^(2*k)) * (1-x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 11 * exp(sqrt(11*n/6)*Pi) / (2^(13/2)*sqrt(3)*n^(3/2)).

A327049 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) (1 + x^(4k)) / ((1 - x^k) (1 - x^(2k)) (1 - x^(3k)) (1 - x^(4*k))).

Original entry on oeis.org

1, 2, 6, 14, 32, 64, 132, 248, 466, 838, 1488, 2560, 4370, 7272, 11988, 19424, 31160, 49280, 77294, 119780, 184164, 280408, 423808, 635136, 945628, 1397398, 2052536, 2995210, 4346416, 6270272, 8999668, 12848584, 18257122, 25817760, 36349600, 50952064, 71131448

Offset: 0

Vaclav Kotesovec, Aug 16 2019

nonn

Convolution of A327046 and A327043.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A015128, A246584, A327048, A327050.

Cf. A301554.

nmax = 50; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k)) * (1+x^(3*k)) * (1+x^(4*k))/((1-x^k) * (1-x^(2*k)) * (1-x^(3*k)) * (1-x^(4*k))), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 5^(5/2) * exp(5*Pi*sqrt(n/3)/2) / (2^(17/2)*3^(3/4)*n^(7/4)).

A327050 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) (1 + x^(4k)) (1 + x^(5k)) / ((1 - x^k) (1 - x^(2k)) (1 - x^(3k)) (1 - x^(4k)) (1 - x^(5*k))).

Original entry on oeis.org

1, 2, 6, 14, 32, 66, 136, 260, 494, 902, 1620, 2832, 4890, 8260, 13792, 22664, 36824, 59060, 93814, 147364, 229490, 354052, 541916, 822736, 1240292, 1856246, 2760368, 4078522, 5990900, 8749052, 12708920, 18363656, 26404386, 37783040, 53820120, 76324576

Offset: 0

Vaclav Kotesovec, Aug 16 2019

nonn

Convolution of A327047 and A327044.

In general, for fixed m>=1, if g.f. = Product_{k>=1} (Product_{j=1..m} (1 + x^(j*k)) / (1 - x^(j*k))), then a(n) ~ sqrt(Gamma(m+1)) * HarmonicNumber(m)^((m+1)/4) * exp(Pi*sqrt(HarmonicNumber(m)*n)) / (2^(3*(m+1)/2) * n^((m+3)/4)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A015128, A246584, A327048, A327049.

Cf. A301554.

nmax = 50; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k)) * (1+x^(3*k)) * (1+x^(4*k)) * (1+x^(5*k)) / ((1-x^k) * (1-x^(2*k)) * (1-x^(3*k)) * (1-x^(4*k)) * (1-x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x]
With[{nn=50,xk=x^(k Range[5])},CoefficientList[Series[Product[Times@@(1+xk)/Times@@(1-xk),{k,nn}],{x,0,nn}],x]] (* Harvey P. Dale, Jul 23 2023 *)

a(n) ~ 137^(3/2) * exp(sqrt(137*n/15)*Pi/2) / (15*2^(21/2)*n^2).

cp's OEIS Frontend

A246585 Zeroth trisection of A246584.

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A246586 First trisection of A246584.

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A246587 Second trisection of A246584.

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A327048 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2*k)) * (1 + x^(3*k)) / ((1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k))).

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A327049 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2*k)) * (1 + x^(3*k)) * (1 + x^(4*k)) / ((1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(4*k))).

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A327050 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2*k)) * (1 + x^(3*k)) * (1 + x^(4*k)) * (1 + x^(5*k)) / ((1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(4*k)) * (1 - x^(5*k))).

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Mathematica

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A327048 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) / ((1 - x^k) (1 - x^(2k)) (1 - x^(3*k))).

A327049 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) (1 + x^(4k)) / ((1 - x^k) (1 - x^(2k)) (1 - x^(3k)) (1 - x^(4*k))).

A327050 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) (1 + x^(4k)) (1 + x^(5k)) / ((1 - x^k) (1 - x^(2k)) (1 - x^(3k)) (1 - x^(4k)) (1 - x^(5*k))).