A327047 -id:A327047 - cp's OEIS frontend

A327045 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3*k)).

1, 1, 2, 4, 5, 8, 13, 17, 24, 36, 47, 64, 89, 115, 152, 204, 260, 336, 438, 552, 702, 896, 1117, 1400, 1758, 2171, 2688, 3332, 4079, 5000, 6131, 7446, 9048, 10992, 13255, 15984, 19264, 23081, 27644, 33084, 39408, 46912, 55797, 66107, 78264, 92572, 109140

Offset: 0

Vaclav Kotesovec, Aug 16 2019

nonn

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

Cf. A000009, A001935, A327046, A327047.

Cf. A107742, A327042.

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

a(n) ~ 11^(1/4) * exp(sqrt(11*n/2)*Pi/3) / (2^(13/4)*sqrt(3)*n^(3/4)).

A327046 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) (1 + x^(4*k)).

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 15, 21, 30, 45, 62, 85, 120, 161, 216, 293, 385, 505, 667, 862, 1112, 1438, 1833, 2330, 2965, 3733, 4688, 5887, 7334, 9114, 11319, 13970, 17203, 21162, 25905, 31643, 38605, 46911, 56891, 68904, 83179, 100224, 120603, 144719, 173360, 207396

Offset: 0

Vaclav Kotesovec, Aug 16 2019

nonn

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

Cf. A000009, A001935, A327045, A327047.

Cf. A107742, A327043.

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

a(n) ~ sqrt(5) * exp(5*Pi*sqrt(n)/6) / (16*sqrt(3)*n^(3/4)).

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

A327045 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3*k)).

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A327046 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) (1 + x^(4*k)).

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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))).

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cp's OEIS Frontend

A327045 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2*k)) * (1 + x^(3*k)).

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A327046 Expansion of Product_{k>=1} (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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A327045 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3*k)).

A327046 Expansion of Product_{k>=1} (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))).