A301554 -id:A301554 - cp's OEIS frontend

A320908 Expansion of Product_{k>=1} theta_4(x^k), where theta_4() is the Jacobi theta function.

1, -2, -2, 2, 4, 6, -6, -2, -8, -12, 2, 6, 20, 14, 22, -2, -14, -34, -20, -42, -48, 34, 10, 50, 48, 80, 82, 52, -16, -30, -142, -130, -138, -226, -54, -70, 80, 190, 310, 238, 392, 178, 178, 86, -40, -148, -582, -506, -546, -680, -656, -126, -336, 262, 428, 930

Offset: 0

Ilya Gutkovskiy, Oct 23 2018

Convolution of A288007 and A288098.

Convolution inverse of A301554.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000005, A000203, A002448, A288007, A288098, A301554, A320067, A320068.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(&*[(1-x^(j*k))/(1+x^(j*k)):j in [1..2*m]]): k in [1..2*m]]) )); // G. C. Greubel, Oct 29 2018

with(numtheory): seq(coeff(series(mul(((1-x^k)/(1+x^k))^tau(k),k=1..n),x,n+1), x, n), n = 0 .. 60); # Muniru A Asiru, Oct 23 2018

nmax = 55; CoefficientList[Series[Product[EllipticTheta[4, 0, x^k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 55; CoefficientList[Series[Product[((1 - x^k)/(1 + x^k))^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 55; CoefficientList[Series[Exp[-Sum[DivisorSigma[1, k] x^k (2 + x^k)/(k (1 - x^(2 k))), {k, 1, nmax}]], {x, 0, nmax}], x]

N=99; x='x+O('x^N); Vec(prod(k=1, N, ((1-x^k)/(1+x^k))^numdiv(k))) \\ Seiichi Manyama, Oct 25 2018

G.f.: Product_{i>=1, j>=1} (1 - x^(i*j))/(1 + x^(i*j)).
G.f.: Product_{k>=1} ((1 - x^k)/(1 + x^k))^d(k), where d(k) is the number of divisors of k (A000005).
G.f.: exp(-Sum_{k>=1} sigma(k)*x^k*(2 + x^k)/(k*(1 - x^(2*k)))).

A305050 Expansion of Product_{i>=1, j>=1, k>=1} (1 + x^(ijk))/(1 - x^(ijk)).

Original entry on oeis.org

1, 2, 8, 20, 56, 128, 316, 684, 1532, 3192, 6704, 13436, 26984, 52352, 101316, 191320, 359334, 662292, 1213360, 2189380, 3925432, 6951592, 12231332, 21298452, 36856840, 63211164, 107765896, 182295468, 306625208, 512190992, 851011960, 1405199028, 2308629300, 3771593392

Offset: 0

Ilya Gutkovskiy, May 24 2018

nonn

Convolution of the sequences A174465 and A280473.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms
Index entries for sequences related to partitions

Cf. A007425, A174465, A280473, A301554.

nmax = 33; CoefficientList[Series[Product[(1 + x^(i j k))/(1 - x^(i j k)), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}], {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^Sum[DivisorSigma[0, d], {d, Divisors[k]}], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A007425(k).

A318814 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(sigma(k)/k).

Original entry on oeis.org

1, 2, 10, 64, 512, 4768, 53056, 645440, 8868352, 133302016, 2184149504, 38530160128, 733246566400, 14834910150656, 319778313883648, 7292507623063552, 175517505539538944, 4440588163825008640, 117969026857318678528, 3276703253565946855424, 95086071773832697348096

Offset: 0

Vaclav Kotesovec, Sep 04 2018

nonn

Convolution of A305127 and A318769.

Vaclav Kotesovec, Table of n, a(n) for n = 0..436

Cf. A000203, A305127, A301554, A301555, A318769.

nmax = 20; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(DivisorSigma[1, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

log(a(n)/n!) ~ Pi^2 * sqrt(n/6).

A318975 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^phi(k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 2, 4, 10, 20, 42, 80, 154, 288, 522, 940, 1658, 2892, 4970, 8456, 14218, 23696, 39122, 64044, 104042, 167732, 268602, 427248, 675482, 1061632, 1659298, 2579676, 3990418, 6142892, 9412906, 14360136, 21814698, 33004704, 49739426, 74677924, 111713658

Offset: 0

Vaclav Kotesovec, Sep 06 2018

nonn

Convolution of A299069 and A061255.

			a(n) ~ exp(3^(4/3) * (7*Zeta(3))^(1/3) * n^(2/3) / (2*Pi^(2/3)) - 1/6) * A^2 * (7*Zeta(3))^(1/9) / (sqrt(2) * 3^(7/18) * Pi^(8/9) * n^(11/18)), where A is the Glaisher-Kinkelin constant A074962.

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

Cf. A061255, A299069, A301554, A301555.

Cf. A318976, A318814, A318977.

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

A318976 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(phi(k)/k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 2, 6, 32, 196, 1512, 13384, 135872, 1545744, 19441952, 268386784, 4018603008, 65021744704, 1127284876928, 20880206388864, 410781080941568, 8561002328678656, 188224613741879808, 4355496092560324096, 105752112730661347328, 2688539359466319184896

Offset: 0

Vaclav Kotesovec, Sep 06 2018

nonn

Convolution of A088009 and A000262.

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Cf. A000262, A088009, A318814, A318977.

Cf. A318975, A301555, A301554.

nmax = 20; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(EulerPhi[k]/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
nmax = 20; CoefficientList[Series[E^(x*(2 + x)/(1 - x^2)), {x, 0, nmax}], x] * Range[0, nmax]!

E.g.f.: exp(x*(2 + x)/(1 - x^2)).

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

A318977 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(tau(k)/k), where tau is A000005.

Original entry on oeis.org

1, 2, 8, 44, 292, 2296, 21472, 221168, 2554544, 32617952, 452957056, 6788855872, 110098330048, 1900192498304, 34971968379392, 683452436531456, 14097619892177152, 306168410773570048, 6998327049216231424, 167369475021548506112, 4187842602663179396096

Offset: 0

Vaclav Kotesovec, Sep 06 2018

nonn

Convolution of A318696 and A318695.

Vaclav Kotesovec, Table of n, a(n) for n = 0..442

Cf. A318695, A318696, A318814, A318976.

Cf. A318975, A301555, A301554.

nmax = 20; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(DivisorSigma[0, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

A320967 Expansion of Product_{k>0} theta_3(q^k)/theta_4(q^k), where theta_3() and theta_4() are the Jacobi theta functions.

Original entry on oeis.org

1, 4, 12, 36, 92, 220, 508, 1108, 2332, 4776, 9492, 18420, 35036, 65324, 119708, 216044, 384204, 674236, 1168968, 2003460, 3397300, 5704148, 9487740, 15642676, 25577900, 41495032, 66817812, 106837112, 169677372, 267755836, 419948980, 654799316, 1015276412, 1565765892

Offset: 0

Seiichi Manyama, Oct 25 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Self-convolution of A320968.

Cf. A000122, A002448, A007096, A301554, A320067, A320970.

With[{nmax=50}, CoefficientList[Series[Product[EllipticTheta[3, 0, q^k]/EllipticTheta[4, 0, q^k], {k, 1, nmax+2}], {q, 0, nmax}], q]] (* G. C. Greubel, Oct 29 2018 *)

m=50; q='q+O('q^m); Vec(prod(k=1,m+2, eta(q^(2*k))^6/(eta(q^k)^4* eta(q^(4*k))^2) )) \\ G. C. Greubel, Oct 29 2018

Expansion of Product_{k>0} eta(q^(2*k))^6 / (eta(q^k)^4*eta(q^(4*k))^2).

A327048 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) / ((1 - x^k) (1 - x^(2k)) (1 - x^(3*k))).

Original entry on oeis.org

1, 2, 6, 14, 30, 60, 120, 220, 402, 710, 1224, 2064, 3438, 5596, 9012, 14304, 22422, 34740, 53330, 80960, 121908, 181976, 269484, 396072, 578232, 838258, 1207896, 1730058, 2463900, 3490020, 4918572, 6897012, 9626610, 13375776, 18504852, 25494456, 34985530

Offset: 0

Vaclav Kotesovec, Aug 16 2019

nonn

Convolution of A327045 and A327042.

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

Cf. A015128, A246584, A327049, A327050.

Cf. A301554.

nmax = 50; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k)) * (1+x^(3*k)) / ((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/6)*Pi) / (2^(13/2)*sqrt(3)*n^(3/2)).

A327049 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) (1 + x^(4k)) / ((1 - x^k) (1 - x^(2k)) (1 - x^(3k)) (1 - x^(4*k))).

Original entry on oeis.org

1, 2, 6, 14, 32, 64, 132, 248, 466, 838, 1488, 2560, 4370, 7272, 11988, 19424, 31160, 49280, 77294, 119780, 184164, 280408, 423808, 635136, 945628, 1397398, 2052536, 2995210, 4346416, 6270272, 8999668, 12848584, 18257122, 25817760, 36349600, 50952064, 71131448

Offset: 0

Vaclav Kotesovec, Aug 16 2019

nonn

Convolution of A327046 and A327043.

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

Cf. A015128, A246584, A327048, A327050.

Cf. A301554.

nmax = 50; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k)) * (1+x^(3*k)) * (1+x^(4*k))/((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/3)/2) / (2^(17/2)*3^(3/4)*n^(7/4)).

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

A320908 Expansion of Product_{k>=1} theta_4(x^k), where theta_4() is the Jacobi theta function.

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A305050 Expansion of Product_{i>=1, j>=1, k>=1} (1 + x^(i*j*k))/(1 - x^(i*j*k)).

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A318814 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(sigma(k)/k).

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A318975 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^phi(k), where phi is the Euler totient function A000010.

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A318976 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(phi(k)/k), where phi is the Euler totient function A000010.

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A318977 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(tau(k)/k), where tau is A000005.

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A320967 Expansion of Product_{k>0} theta_3(q^k)/theta_4(q^k), where theta_3() and theta_4() are the Jacobi theta functions.

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A327048 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2*k)) * (1 + x^(3*k)) / ((1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k))).

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A305050 Expansion of Product_{i>=1, j>=1, k>=1} (1 + x^(ijk))/(1 - x^(ijk)).

A327048 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) / ((1 - x^k) (1 - x^(2k)) (1 - x^(3*k))).

A327049 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) (1 + x^(4k)) / ((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))).