A156616 -id:A156616 - cp's OEIS frontend

A261520 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(3^k).

1, 6, 36, 200, 1038, 5160, 24776, 115632, 527172, 2355998, 10349448, 44783064, 191211512, 806737800, 3367294320, 13918479872, 57020736942, 231697484304, 934399998412, 3742041461976, 14888854356840, 58881590423856, 231542984619720, 905666813058384

Offset: 0

Vaclav Kotesovec, Aug 23 2015

nonn

Convolution of A144067 and A256142.

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

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO]
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, p. 27.

Cf. A156616, A261519, A260916, A001934, A015128.

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

a(n) ~ 3^n * exp(2*sqrt(2*n) - 1 + c) / (sqrt(Pi) * 2^(3/4) * n^(3/4)), where c = 2 * Sum_{j>=1} 1/((2*j+1)*(3^(2*j)-1)) = 0.0887630729103166089354170592729856346...

A285447 Expansion of Product_{k>=1} ((1 + x^(3*k)) / (1 - x^k))^k.

Original entry on oeis.org

1, 1, 3, 7, 14, 27, 56, 101, 190, 347, 617, 1082, 1895, 3230, 5490, 9226, 15332, 25259, 41356, 67021, 107989, 172789, 274613, 433815, 681650, 1064661, 1654739, 2559029, 3938438, 6033967, 9205152, 13982675, 21156174, 31886290, 47879210, 71636483, 106814323

Offset: 0

Vaclav Kotesovec, Apr 19 2017

nonn

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

Cf. A266648, A285446.

Cf. A156616, A285462, A285460, A285461.

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

a(n) ~ exp(1/12 + 2^(-4/3) * 3^(2/3) * (13*Zeta(3))^(1/3) * n^(2/3)) * (13*Zeta(3))^(7/36) / (A * 2^(7/9) * 3^(25/36) * sqrt(Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.

A261389 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(3*k).

Original entry on oeis.org

1, 6, 30, 128, 486, 1704, 5604, 17484, 52206, 150118, 417696, 1128984, 2973476, 7650720, 19272432, 47616568, 115570014, 275921460, 648771802, 1503889488, 3439990344, 7770915816, 17349229908, 38306180052, 83694778556, 181052778078, 387976101432, 823939048560

Offset: 0

Vaclav Kotesovec, Aug 17 2015

nonn

Convolution of A255610 and A027346.

In general, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(t*k) and t>=1, then a(n) ~ exp(t/12 + 3/2 * (7*t*Zeta(3)/2)^(1/3) * n^(2/3)) * t^(1/6 + t/36) * (7*Zeta(3))^(1/6 + t/36) / (A^t * 2^(2/3 + t/9) * sqrt(3*Pi) * n^(2/3 + t/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

G. C. Greubel, 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, p. 19.

Cf. A156616 (t=1), A261386 (t=2).

Cf. A015128, A027346, A027906, A193427, A255610.

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

a(n) ~ exp(1/4 + 3/2 * (21*Zeta(3)/2)^(1/3) * n^(2/3)) * (7*Zeta(3)/3)^(1/4) / (2 * A^3 * sqrt(Pi) * n^(3/4)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A261452 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(2*k-1).

Original entry on oeis.org

1, 2, 8, 24, 66, 176, 448, 1096, 2608, 6042, 13664, 30280, 65856, 140800, 296432, 615264, 1260306, 2550368, 5102616, 10101000, 19797344, 38439088, 73976160, 141179480, 267300752, 502283714, 937077808, 1736296304, 3196144032, 5846632656, 10631038400

Offset: 0

Vaclav Kotesovec, Aug 19 2015

nonn

Convolution of A253289 and A255835.

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, p. 26.

Cf. A156616, A261386, A005309, A261451, A261389.

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 2^(1/3) * (7*Zeta(3))^(1/18) * exp(1/6 - Pi^4/(672*Zeta(3)) - Pi^2 * n^(1/3)/(4*(7*Zeta(3))^(1/3)) + 3/2*(7*Zeta(3))^(1/3) * n^(2/3)) / (A^2 * sqrt(3) * n^(5/9)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A285446 Expansion of Product_{k>=1} ((1 + x^k) / (1 - x^(3*k)))^k.

Original entry on oeis.org

1, 1, 2, 6, 9, 18, 36, 60, 105, 191, 314, 528, 896, 1447, 2355, 3831, 6071, 9619, 15207, 23648, 36693, 56724, 86762, 132264, 200853, 302699, 454565, 680061, 1011540, 1499363, 2214570, 3255796, 4770830, 6967967, 10137577, 14703909, 21262751, 30644816, 44041843

Offset: 0

Vaclav Kotesovec, Apr 19 2017

nonn

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

Cf. A285445, A285447.

Cf. A156616, A000219, A285458, A285459.

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

a(n) ~ exp(1/12 + 2^(-4/3) * (93*Zeta(3))^(1/3) * n^(2/3)) * (31*Zeta(3))^(7/36) / (A * 2^(7/9) * 3^(29/36) * sqrt(Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.

A285460 Expansion of Product_{k>=1} ((1 + x^(4*k)) / (1 - x^k))^k.

Original entry on oeis.org

1, 1, 3, 6, 14, 25, 51, 92, 175, 308, 554, 957, 1670, 2820, 4778, 7940, 13169, 21511, 35032, 56405, 90453, 143716, 227342, 356950, 557977, 866588, 1340109, 2060912, 3156274, 4810016, 7301490, 11034661, 16614681, 24916208, 37234864, 55440054, 82274277

Offset: 0

Vaclav Kotesovec, Apr 19 2017

nonn

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

Cf. A156616, A285462, A285447, A285461, A285472.

nmax = 40; CoefficientList[Series[Product[((1+x^(4*k))/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(1/12 + 3 * 2^(-8/3) * (67*Zeta(3))^(1/3) * n^(2/3)) * (67*Zeta(3))^(7/36) / (A * 2^(14/9) * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.

A295792 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(1/k).

Original entry on oeis.org

1, 2, 6, 28, 152, 1008, 7936, 70208, 689664, 7618816, 92013824, 1202362368, 17053410304, 258928934912, 4197838491648, 72840915607552, 1334630802489344, 25799982480556032, 527187369241870336, 11292834065764450304, 253498950169144590336, 5965951790211865772032, 146341359815078034538496

Offset: 0

Ilya Gutkovskiy, Nov 27 2017

nonn

Convolution of A028342 and A168243. - Vaclav Kotesovec, Sep 07 2018

Vaclav Kotesovec, Table of n, a(n) for n = 0..445

Cf. A001227, A028342, A028343, A054844, A156616, A168243, A206303, A294356.

a:=series(mul(((1+x^k)/(1-x^k))^(1/k),k=1..100),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 27 2019

nmax = 22; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: exp(2*Sum_{k>=1} A001227(k)*x^k/k).

E.g.f.: exp(Sum_{k>=1} A054844(k)*x^k/k).

A261563 Expansion of Product_{k>=1} ((1 + 2x^k)/(1 - 2x^k))^k.

Original entry on oeis.org

1, 4, 16, 60, 192, 596, 1744, 4892, 13248, 34868, 89296, 223660, 548928, 1323060, 3137520, 7332332, 16907584, 38517444, 86777328, 193523404, 427562816, 936555044, 2035286576, 4390850268, 9409096576, 20037827876, 42429318480, 89369282460, 187325508288

Offset: 0

Vaclav Kotesovec, Aug 24 2015

nonn

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

Cf. A156616, A261561, A261562, A261584.

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

{a(n) = polcoeff( exp( sum(m=1, n, x^m/m * sumdiv(m, d, (2^d - (-2)^d) * m^2/d^2) ) +x*O(x^n)), n)}
for(n=0,40,print1(a(n),", ")) \\ Paul D. Hanna, Sep 30 2015

a(n) ~ c * 2^n, where c = 2 * Product_{j>=1} ((1 + 1/2^j)/(1 - 1/2^j))^(j+1) = 1021.5383556752320172813996404366861329314041364322798995039038143325883...

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} (2^d - (-2)^d) * n^2/d^2 ). - Paul D. Hanna, Sep 30 2015

A285461 Expansion of Product_{k>=1} ((1 + x^(5*k)) / (1 - x^k))^k.

Original entry on oeis.org

1, 1, 3, 6, 13, 25, 49, 89, 166, 295, 526, 909, 1571, 2657, 4475, 7432, 12257, 20000, 32436, 52126, 83285, 132057, 208221, 326202, 508372, 787777, 1214828, 1863932, 2847020, 4328765, 6554359, 9882795, 14843999, 22210386, 33112817, 49192218, 72834243

Offset: 0

Vaclav Kotesovec, Apr 19 2017

nonn

In general, if m >= 1 and g.f. = Product_{k>=1} ((1 + x^(m*k)) / (1 - x^k))^k, then a(n, m) ~ exp(1/12 + 3 * 2^(-4/3) * (4 + 3/m^2)^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * (4 + 3/m^2)^(7/36) * Zeta(3)^(7/36) / (A * 2^(7/9) * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.

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

Cf. A156616 (m=1), A285462 (m=2), A285447 (m=3), A285460 (m=4).

Cf. A024789.

nmax = 40; CoefficientList[Series[Product[((1+x^(5*k))/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(1/12 + 3 * 2^(-4/3) * 5^(-2/3) * (103*Zeta(3))^(1/3) * n^(2/3)) * (103*Zeta(3))^(7/36) / (A * 2^(7/9) * 5^(7/18) * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.

A285462 Expansion of Product_{k>=1} ((1 + x^(2*k)) / (1 - x^k))^k.

Original entry on oeis.org

1, 1, 4, 7, 18, 32, 72, 127, 257, 454, 861, 1497, 2719, 4654, 8171, 13781, 23564, 39159, 65559, 107455, 176712, 286000, 463200, 740910, 1184123, 1873656, 2959376, 4636145, 7245680, 11246590, 17409731, 26792371, 41114202, 62769820, 95553779, 144803917

Offset: 0

Vaclav Kotesovec, Apr 19 2017

nonn

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

Cf. A156616, A285447, A285460, A285461.

Cf. A279328.

nmax = 40; CoefficientList[Series[Product[((1+x^(2*k))/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(1/12 + 3 * (19*Zeta(3))^(1/3) * n^(2/3) / 4) * (19*Zeta(3))^(7/36) / (A * 2^(7/6) * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.

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A261520 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(3^k).

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A285447 Expansion of Product_{k>=1} ((1 + x^(3*k)) / (1 - x^k))^k.

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A261389 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(3*k).

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A261452 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(2*k-1).

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A285446 Expansion of Product_{k>=1} ((1 + x^k) / (1 - x^(3*k)))^k.

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A285460 Expansion of Product_{k>=1} ((1 + x^(4*k)) / (1 - x^k))^k.

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A295792 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(1/k).

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A261563 Expansion of Product_{k>=1} ((1 + 2*x^k)/(1 - 2*x^k))^k.

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A285461 Expansion of Product_{k>=1} ((1 + x^(5*k)) / (1 - x^k))^k.

A261563 Expansion of Product_{k>=1} ((1 + 2x^k)/(1 - 2x^k))^k.