A270919 -id:A270919 - cp's OEIS frontend

A361008 G.f.: Product_{k >= 0} ((1 + x^(2k+1)) / (1 - x^(2k+1)))^k.

1, 0, 0, 2, 0, 4, 2, 6, 8, 10, 20, 18, 42, 40, 78, 92, 140, 192, 258, 382, 480, 728, 902, 1334, 1698, 2404, 3148, 4292, 5742, 7608, 10304, 13430, 18192, 23592, 31720, 41144, 54766, 71188, 93762, 122156, 159420, 207820, 269380, 350726, 452434, 587520, 755446

Offset: 0

Vaclav Kotesovec, Apr 20 2023

nonn

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

Cf. A080054, A291697, A263149, A263150.

Cf. A156616, A270919.

Table[SeriesCoefficient[Product[((1 + x^(2*k + 1))/(1 - x^(2*k + 1)))^k, {k, 0, n}], {x, 0, n}], {n, 0, 50}]

a(n) ~ sqrt(A/(3*Pi)) * (7*zeta(3))^(11/72) * exp(3*(7*zeta(3))^(1/3) * n^(2/3)/4 - Pi^2 * n^(1/3)/(8*(7*zeta(3))^(1/3)) - 1/24 - Pi^4/(1344*zeta(3))) / (2^(3/4) * n^(47/72)), where A = A074962 is the Glaisher-Kinkelin constant.

A294592 a(n) = [x^n] (theta_3(x)/theta_4(x))^n, where theta_() is the Jacobi theta function.

Original entry on oeis.org

1, 4, 32, 304, 3072, 32024, 340352, 3666016, 39878656, 437091892, 4819567552, 53401892240, 594093969408, 6631726263608, 74242911364864, 833237193123104, 9371924860764160, 105614054423502408, 1192210691317862048, 13478559927485340144, 152589996020498655232, 1729590806617202662528

Offset: 0

Ilya Gutkovskiy, Nov 03 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..500
Eric Weisstein's MathWorld, Jacobi Theta Functions
Wikipedia, Theta function

Cf. A007096, A014969, A014970, A014972, A066535, A080054, A103261, A270919, A289522, A291697.

S:= series((JacobiTheta3(0,x)/JacobiTheta4(0,x))^n,x,51):
seq(coeff(S,x,n),n=0..50); # Robert Israel, Nov 03 2017

Table[SeriesCoefficient[(EllipticTheta[3, 0, x]/EllipticTheta[4, 0, x])^n, {x, 0, n}], {n, 0, 21}]
Table[SeriesCoefficient[Product[((1 + x^(2 k + 1))/(1 - x^(2 k + 1)))^(2 n), {k, 0, n}], {x, 0, n}], {n, 0, 21}]
Table[SeriesCoefficient[(QPochhammer[-x, x^2]/QPochhammer[x, x^2])^(2 n), {x, 0, n}], {n, 0, 21}]
(* Calculation of constant d: *) 1/r /. FindRoot[{s == EllipticTheta[3, 0, r*s]/EllipticTheta[4, 0, r*s], EllipticTheta[4, 0, r*s] + r*s*Derivative[0, 0, 1][EllipticTheta][4, 0, r*s] == r*Derivative[0, 0, 1][EllipticTheta][3, 0, r*s]}, {r, 1/10}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Jan 17 2024 *)

a(n) = [x^n] Product_{k>=0} ((1 + x^(2*k+1))/(1 - x^(2*k+1)))^(2*n).
From Vaclav Kotesovec, Nov 05 2017: (Start)
a(n) ~ c * d^n / sqrt(n), where
d = 11.61255065799699699891360038489317237925475956178123836149123386457... and
c = 0.34456510029264878768512693687607064416428117641473856418257649837... (End)

A319457 a(n) = [x^n] Product_{k>=1} 1/((1 - x^k)(1 - x^(2k)))^n.

Original entry on oeis.org

1, 1, 7, 31, 175, 931, 5209, 29114, 165087, 940828, 5396777, 31090962, 179832625, 1043516371, 6072302726, 35420582431, 207051636799, 1212583329959, 7113193757656, 41788933655049, 245831162935825, 1447891754747672, 8537111315442222, 50387162650271055, 297664212003582753

Offset: 0

Ilya Gutkovskiy, Sep 19 2018

nonn

Cf. A002513, A008485, A270919, A296043, A296044, A319455, A319456.

Table[SeriesCoefficient[Product[1/((1 - x^k) (1 - x^(2 k)))^n , {k, 1, n}], {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[1/(QPochhammer[x] QPochhammer[x^2])^n, {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[Exp[n Sum[(4 DivisorSigma[1, k] - DivisorSigma[1, 2 k]) x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 24}]

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

a(n) = [x^n] exp(n*Sum_{k>=1} (4*sigma(k) - sigma(2*k))*x^k/k).

A319672 a(n) = [x^n] Product_{k>=2} ((1 + x^k)/(1 - x^k))^n.

Original entry on oeis.org

1, 0, 4, 6, 40, 110, 520, 1778, 7568, 28320, 116224, 453046, 1837600, 7306234, 29565848, 118786526, 481192480, 1945153838, 7895908852, 32046260282, 130370798320, 530650047710, 2163191769336, 8824509524082, 36037768384832, 147277910160160, 602398740105712, 2465582764631334

Offset: 0

Ilya Gutkovskiy, Sep 25 2018

nonn

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

Cf. A000203, A270919, A300415, A319670, A319671.

Table[SeriesCoefficient[Product[((1 + x^k)/(1 - x^k))^n , {k, 2, n}], {x, 0, n}], {n, 0, 27}]
Table[SeriesCoefficient[((1 - x)/((1 + x) EllipticTheta[4, 0, x]))^n, {x, 0, n}], {n, 0, 27}]
Table[SeriesCoefficient[Exp[n Sum[(DivisorSigma[1, 2 k] - DivisorSigma[1, k] + (-1)^k - 1) x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 27}]

a(n) = [x^n] ((1 - x)/((1 + x)*theta_4(x)))^n, where theta_4() is the Jacobi theta function.
a(n) = [x^n] exp(n*Sum_{k>=1} (sigma(2*k) - sigma(k) + (-1)^k - 1)*x^k/k).
a(n) ~ c * d^n / sqrt(n), where d = 4.16958962845360844086951404338054148667024... and c = 0.23380422010834870751549442953816486722... - Vaclav Kotesovec, Oct 06 2018

A300412 a(n) = [x^n] Product_{k>=1} ((1 + nx^k)/(1 - nx^k))^k.

Original entry on oeis.org

1, 2, 16, 144, 1376, 15800, 210816, 3333372, 61688448, 1318588146, 32004369200, 869282342632, 26099925704928, 857736429098848, 30605729417479104, 1177841009504482200, 48614265201514729984, 2141639401723095243324, 100282931820560447963568, 4973060138191518242569120

Offset: 0

Ilya Gutkovskiy, Mar 05 2018

nonn

			The table of coefficients of x^k in expansion of Product_{k>=1} ((1 + n*x^k)/(1 - n*x^k))^k begins:
n = 0: (1),  0,   0,    0,     0,       0,  ...
n = 1:  1,  (2),  6,   16,    38,      88,  ...
n = 2:  1,   4, (16),  60,   192,     596,  ...
n = 3:  1,   6,  30, (144),  582,    2280,  ...
n = 4:  1,   8,  48,  280, (1376),   6568,  ...
n = 5:  1,  10,  70,  480,  2790,  (15800), ...

Cf. A156616, A261563, A266942, A270919, A270924, A292419, A298985, A298987.

Table[SeriesCoefficient[Product[((1 + n x^k)/(1 - n x^k))^k, {k, 1, n}], {x, 0, n}], {n, 0, 19}]

a(n) ~ 2 * n^n * (1 + 4/n + 14/n^2 + 44/n^3 + 124/n^4 + 328/n^5 + 824/n^6 + 1980/n^7 + 4590/n^8 + 10320/n^9 + 22584/n^10 + ...), for coefficients see A261451. - Vaclav Kotesovec, Mar 05 2018

A303483 a(n) = [x^n] Product_{k=1..n} ((1 + x^k)/(1 - x^k))^(n-k+1).

Original entry on oeis.org

1, 2, 10, 64, 436, 3072, 22096, 161148, 1187118, 8812050, 65806720, 493827256, 3720698056, 28128081912, 213258301824, 1620878656280, 12346263051028, 94221026620572, 720267101230410, 5514346833878672, 42274910234115352, 324490877248800232, 2493471670778297856, 19179885230907692452

Offset: 0

Ilya Gutkovskiy, Apr 24 2018

nonn

			a(0) = 1;
a(1) = [x^1] (1 + x)/(1 - x) = 2;
a(2) = [x^2] ((1 + x)^2*(1 + x^2))/((1 - x)^2*(1 - x^2)) = 10;
a(3) = [x^3] ((1 + x)^3*(1 + x^2)^2*(1 + x^3))/((1 - x)^3*(1 - x^2)^2*(1 - x^3)) = 64;
a(4) = [x^4] ((1 + x)^4*(1 + x^2)^3*(1 + x^3)^2*(1 + x^4))/((1 - x)^4*(1 - x^2)^3*(1 - x^3)^2*(1 - x^4)) = 436;
a(5) = [x^5] ((1 + x)^5*(1 + x^2)^4*(1 + x^3)^3*(1 + x^4)^2*(1 + x^5))/((1 - x)^5*(1 - x^2)^4*(1 - x^3)^3*(1 - x^4)^2*(1 - x^5)) = 3072, etc.
...
The table of coefficients of x^k in expansion of Product_{k=1..n} ((1 + x^k)/(1 - x^k))^(n-k+1) begins:
n = 0: (1),  0,   0,    0,    0,     0,  ...
n = 1:  1,  (2),  2,    2,    2,     2,  ...
n = 2:  1,   4, (10),  20,   34,    52,  ...
n = 3:  1,   6,  22,  (64), 158,   346,  ...
n = 4:  1,   8,  38,  140, (436), 1200,  ...
n = 5:  1,  10,  58,  256,  946, (3072), ...

Cf. A206228, A206229, A270919, A303173, A303174.

Table[SeriesCoefficient[Product[((1 + x^k)/(1 - x^k))^(n - k + 1), {k, 1, n}], {x, 0, n}], {n, 0, 23}]

a(n) ~ c * d^n / sqrt(n), where d = 7.862983395705905261519347909953827161057584... and c = 0.23317816342157644853479309078... - Vaclav Kotesovec, May 04 2018

A304461 Coefficient of x^n in Product_{k>=1} ((1+x^k)/(1-x^k))^(n^3).

Original entry on oeis.org

1, 2, 144, 29232, 12263552, 8807437800, 9671073636672, 15075101792958592, 31660212257148109824, 86182291753025176234602, 295133367252867736074882400, 1241742977667269060006125955952, 6296492342467004634980003629748736, 37869525230334631809014462278624137096

Offset: 0

Vaclav Kotesovec, May 13 2018

nonn

In general, for m>=3, coefficient of x^n in Product_{k>=1} ((1+x^k)/(1-x^k))^(n^m) is asymptotic to 2^n * n^(m*n) / n!.

Cf. A206623, A270919, A304448, A304459, A304460.

nmax = 20; Table[SeriesCoefficient[Product[((1+x^k)/(1-x^k))^(n^3), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 20; Table[SeriesCoefficient[(QPochhammer[-1, x]/2/QPochhammer[x])^(n^3), {x, 0, n}], {n, 0, nmax}]

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

cp's OEIS Frontend

A361008 G.f.: Product_{k >= 0} ((1 + x^(2*k+1)) / (1 - x^(2*k+1)))^k.

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A294592 a(n) = [x^n] (theta_3(x)/theta_4(x))^n, where theta_() is the Jacobi theta function.

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A319457 a(n) = [x^n] Product_{k>=1} 1/((1 - x^k)*(1 - x^(2*k)))^n.

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A319672 a(n) = [x^n] Product_{k>=2} ((1 + x^k)/(1 - x^k))^n.

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A300412 a(n) = [x^n] Product_{k>=1} ((1 + n*x^k)/(1 - n*x^k))^k.

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A303483 a(n) = [x^n] Product_{k=1..n} ((1 + x^k)/(1 - x^k))^(n-k+1).

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A304461 Coefficient of x^n in Product_{k>=1} ((1+x^k)/(1-x^k))^(n^3).

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A361008 G.f.: Product_{k >= 0} ((1 + x^(2k+1)) / (1 - x^(2k+1)))^k.

A319457 a(n) = [x^n] Product_{k>=1} 1/((1 - x^k)(1 - x^(2k)))^n.

A300412 a(n) = [x^n] Product_{k>=1} ((1 + nx^k)/(1 - nx^k))^k.