A285291 -id:A285291 - cp's OEIS frontend

A263345 Expansion of Product_{k>=1} ((1 + x^k)/(1 + x^(3*k)))^k.

1, 1, 2, 4, 7, 14, 22, 40, 65, 107, 176, 282, 448, 705, 1101, 1701, 2611, 3977, 6021, 9048, 13527, 20102, 29720, 43712, 63997, 93259, 135317, 195539, 281440, 403559, 576568, 820888, 1164826, 1647583, 2323169, 3266041, 4578305, 6399990, 8922389, 12406535

Offset: 0

Vaclav Kotesovec, Oct 15 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A003105, A026007, A262876, A262877, A262878, A262879, A263346.

Cf. A285289, A285290, A285291.

nmax=40; CoefficientList[Series[Product[((1 + x^k)/(1 + x^(3*k)))^k,{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ Zeta(3)^(1/6) * exp(2^(-1/3) * 3^(2/3) * Zeta(3)^(1/3) * n^(2/3)) / (2^(1/6) * 3^(2/3) * sqrt(Pi) * n^(2/3)).

A285290 Expansion of Product_{k>=1} ((1 + x^k) / (1 + x^(4*k)))^k.

Original entry on oeis.org

1, 1, 2, 5, 7, 15, 26, 44, 74, 125, 205, 331, 534, 844, 1332, 2077, 3215, 4934, 7533, 11410, 17191, 25751, 38346, 56833, 83814, 123025, 179776, 261639, 379186, 547476, 787516, 1128775, 1612395, 2295701, 3258177, 4610130, 6503873, 9149365, 12835612, 17959085

Offset: 0

Vaclav Kotesovec, Apr 16 2017

nonn

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

Cf. A285289, A263345, A285291.

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

a(n) ~ exp(2^(-8/3) * 3^(5/3) * (5*Zeta(3))^(1/3) * n^(2/3)) * (5*Zeta(3))^(1/6) / (2^(4/3) * 3^(1/6) * sqrt(Pi) * n^(2/3)).

A285289 Expansion of Product_{k>=1} ((1 + x^k) / (1 + x^(2*k)))^k.

Original entry on oeis.org

1, 1, 1, 4, 5, 10, 16, 26, 44, 68, 110, 167, 265, 399, 609, 919, 1371, 2040, 3005, 4420, 6436, 9364, 13501, 19433, 27806, 39639, 56265, 79572, 112126, 157390, 220283, 307163, 427145, 592029, 818359, 1127878, 1550483, 2125656, 2907013, 3965853, 5397497

Offset: 0

Vaclav Kotesovec, Apr 16 2017

nonn

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

Cf. A263345, A285290, A285291.

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

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

A304629 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(5*k)))^n.

Original entry on oeis.org

1, 1, 3, 13, 51, 201, 819, 3389, 14131, 59341, 250703, 1064207, 4535091, 19390229, 83139955, 357354213, 1539272499, 6642769925, 28714955571, 124312591469, 538895612751, 2338948779320, 10162837993377, 44202371860240, 192431323820851, 838442649862701, 3656031108325651

Offset: 0

Ilya Gutkovskiy, May 15 2018

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A096938, A255526, A285291, A296163, A296164, A304628.

Table[SeriesCoefficient[Product[((1 + x^k)/(1 + x^(5 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[Product[1/(1 - x^k + x^(2 k) - x^(3 k) + x^(4 k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
(* Calculation of constants {d,c}: *) With[{k = 5}, {1/r, Sqrt[QPochhammer[-1, (r*s)^k] / (2*Pi*(r^2*s*Derivative[0, 2][QPochhammer][-1, r*s] - k^2*(r*s)^(2*k) * Derivative[0, 2][QPochhammer][-1, (r*s)^k] - k*(1 + k)*(r*s)^k * Derivative[0, 1][QPochhammer][-1, (r*s)^k]))]} /. FindRoot[{s == QPochhammer[-1, r*s] / QPochhammer[-1, (r*s)^k], QPochhammer[-1, (r*s)^k] + k*(r*s)^k*Derivative[0, 1][QPochhammer][-1, (r*s)^k] == r*Derivative[0, 1][QPochhammer][-1, r*s]}, {r, 1/4}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

a(n) = [x^n] Product_{k>=1} 1/(1 - x^k + x^(2*k) - x^(3*k) + x^(4*k))^n.

a(n) ~ c * d^n / sqrt(n), where d = 4.445766346387064439086120427... and c = 0.267035948020079842478289... - Vaclav Kotesovec, May 18 2018

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A263345 Expansion of Product_{k>=1} ((1 + x^k)/(1 + x^(3*k)))^k.

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A285290 Expansion of Product_{k>=1} ((1 + x^k) / (1 + x^(4*k)))^k.

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A285289 Expansion of Product_{k>=1} ((1 + x^k) / (1 + x^(2*k)))^k.

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A304629 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(5*k)))^n.

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