A327044 -id:A327044 - cp's OEIS frontend

A327042 Expansion of Product_{k>=1} 1/((1 - x^k) * (1 - x^(2k)) (1 - x^(3*k))).

1, 1, 3, 5, 10, 15, 29, 42, 72, 107, 170, 246, 382, 541, 807, 1139, 1650, 2292, 3267, 4479, 6261, 8518, 11716, 15771, 21449, 28599, 38430, 50876, 67654, 88854, 117171, 152775, 199785, 258901, 336024, 432744, 558027, 714494, 915555, 1166243, 1485792, 1883031

Offset: 0

Vaclav Kotesovec, Aug 16 2019

nonn

Differs from A006168.

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

Cf. A000041, A002513, A327043, A327044.

Cf. A006171.

nmax = 50; CoefficientList[Series[Product[1/((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)*Pi/3) / (48*sqrt(3)*n^(3/2)).

A327043 Expansion of Product_{k>=1} 1/((1 - x^k) * (1 - x^(2k)) (1 - x^(3k)) (1 - x^(4*k))).

Original entry on oeis.org

1, 1, 3, 5, 11, 16, 32, 47, 84, 124, 205, 298, 477, 681, 1044, 1484, 2211, 3097, 4516, 6261, 8948, 12295, 17273, 23511, 32597, 43975, 60187, 80601, 109114, 144999, 194423, 256584, 341008, 447178, 589558, 768398, 1005854, 1303450, 1694815, 2184666, 2823229

Offset: 0

Vaclav Kotesovec, Aug 16 2019

nonn

Differs from A006169.

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

Cf. A000041, A002513, A327042, A327044.

Cf. A006171.

nmax = 50; CoefficientList[Series[Product[1/((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/2)/3) / (288*2^(1/4)*n^(7/4)).

A327047 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) (1 + x^(4k)) (1 + x^(5*k)).

Original entry on oeis.org

1, 1, 2, 4, 6, 10, 16, 23, 34, 51, 72, 101, 143, 195, 267, 366, 487, 650, 866, 1135, 1487, 1940, 2504, 3226, 4145, 5283, 6714, 8513, 10725, 13481, 16905, 21085, 26244, 32588, 40299, 49732, 61229, 75131, 92004, 112435, 137009, 166627, 202269, 244919, 296038

Offset: 0

Vaclav Kotesovec, Aug 16 2019

nonn

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

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

Cf. A000009, A001935, A327045, A327046.

Cf. A107742, A327044.

nmax = 50; CoefficientList[Series[Product[(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]

a(n) ~ 137^(1/4) * exp(sqrt(137*n/5)*Pi/6) / (2^(9/2)*sqrt(3)*5^(1/4)*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

A327042 Expansion of Product_{k>=1} 1/((1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k))).

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A327043 Expansion of Product_{k>=1} 1/((1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(4*k))).

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A327047 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2*k)) * (1 + x^(3*k)) * (1 + x^(4*k)) * (1 + x^(5*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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A327042 Expansion of Product_{k>=1} 1/((1 - x^k) * (1 - x^(2k)) (1 - x^(3*k))).

A327043 Expansion of Product_{k>=1} 1/((1 - x^k) * (1 - x^(2k)) (1 - x^(3k)) (1 - x^(4*k))).

A327047 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) (1 + x^(4k)) (1 + x^(5*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))).