A301554 -id:A301554 - cp's OEIS frontend

A302237 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k*(k+1)/2).

1, 2, 8, 26, 76, 216, 590, 1554, 3988, 9988, 24464, 58794, 138866, 322808, 739658, 1672372, 3734848, 8245956, 18012114, 38952586, 83448832, 177194716, 373111970, 779430870, 1615995262, 3326484686, 6800794428, 13813260736, 27881653590, 55942340000, 111601021856

Offset: 0

Ilya Gutkovskiy, Apr 03 2018

nonn

Convolution of the sequences A000294 and A028377.

N. J. A. Sloane, Transforms

Cf. A000217, A000294, A015128, A028377, A156616, A206622, A206623, A206624, A260916, A261386, A261452, A261519, A261520, A301554, A301555.

nmax = 30; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A000217(k).
a(n) ~ exp(2*Pi*n^(3/4)/3 + 7*Zeta(3)*sqrt(n) / (2*Pi^2) - 49*Zeta(3)^2 * n^(1/4) / (4*Pi^5) + 22411 * Zeta(3)^3 / (392*Pi^8) - Zeta(3)/(8*Pi^2) + 1/24) * Pi^(1/24) / (sqrt(A) * 2^(25/12) * n^(61/96)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 08 2018
G.f.: A(x) = exp( 2*Sum_{n >= 0} x^(2*n+1)/((2*n+1)*(1 - x^(2*n+1))^3) ). Cf. A000122 and A156616. - Peter Bala, Dec 23 2021

A308286 Expansion of Product_{i>=1, j>=1} theta_3(x^(i*j)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 4, 12, 20, 40, 84, 140, 252, 456, 752, 1260, 2128, 3392, 5436, 8760, 13582, 21092, 32744, 49620, 75104, 113448, 168508, 249620, 368840, 538412, 783480, 1136652, 1634000, 2341280, 3344680, 4743684, 6706120, 9452392, 13245800, 18504888, 25777520, 35735376

Offset: 0

Ilya Gutkovskiy, May 18 2019

nonn

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

Cf. A000005, A000122, A006171, A007425, A300446, A301554, A305050, A308288, A320067, A320908, A321241.

nmax = 37; CoefficientList[Series[Product[Product[EllipticTheta[3, 0, x^(i j)], {j, 1, nmax}], {i, 1, nmax}], {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[Product[EllipticTheta[3, 0, x^k]^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} theta_3(x^k)^tau(k), where tau = number of divisors (A000005).
G.f.: Product_{i>=1, j>=1} (Sum_{k=-oo..+oo} x^(i*j*k^2)).
G.f.: Product_{i>=1, j>=1, k>=1} (1 - x^(i*j*k))*(1 + x^(i*j*k))^3/(1 + x^(2*i*j*k))^2.
G.f.: Product_{k>=1} (1 - x^k)^tau_3(k)*(1 + x^k)^(3*tau_3(k))/(1 + x^(2*k))^(2*tau_3(k)), where tau_3 = A007425.

A308288 Expansion of Product_{i>=1, j>=1} theta_3(x^(ij))/theta_4(x^(ij)), where theta_() is the Jacobi theta function.

Original entry on oeis.org

1, 4, 16, 56, 172, 496, 1360, 3528, 8824, 21344, 50048, 114360, 255336, 557888, 1195952, 2519264, 5221076, 10660512, 21467904, 42674520, 83812560, 162753584, 312689168, 594740456, 1120498048, 2092059800, 3872731232, 7110830376, 12955269304, 23428775520

Offset: 0

Ilya Gutkovskiy, May 18 2019

nonn

Convolution of the sequences A305050 and A308286.

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

Cf. A000005, A000122, A002448, A007096, A007425, A301554, A305050, A308286, A320967, A320970.

nmax = 29; CoefficientList[Series[Product[Product[EllipticTheta[3, 0, x^(i j)]/EllipticTheta[4, 0, x^(i j)], {j, 1, nmax}], {i, 1, nmax}], {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[Product[(EllipticTheta[3, 0, x^k]/EllipticTheta[4, 0, x^k])^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (theta_3(x^k)/theta_4(x^k))^tau(k), where tau = number of divisors (A000005).
G.f.: Product_{i>=1, j>=1} (Sum_{k=-oo..+oo} x^(i*j*k^2))/(Sum_{k=-oo..+oo} (-1)^k*x^(i*j*k^2)).
G.f.: Product_{i>=1, j>=1, k>=1} (1 + x^(i*j*k))^4/(1 + x^(2*i*j*k))^2.
G.f.: Product_{k>=1} (1 + x^k)^(4*tau_3(k))/(1 + x^(2*k))^(2*tau_3(k)), where tau_3 = A007425.

A318579 Expansion of Product_{i>=1, j>=1} ((1 + x^(ij))/(1 - x^(ij)))^(i*j).

Original entry on oeis.org

1, 2, 10, 30, 98, 270, 786, 2046, 5418, 13556, 33726, 81002, 192902, 447562, 1027990, 2316750, 5165398, 11345298, 24668952, 52972902, 112688802, 237193354, 494933514, 1023238806, 2098662698, 4269141516, 8620916966, 17280687472, 34405835066, 68044209950, 133732805458

Offset: 0

Ilya Gutkovskiy, Aug 29 2018

nonn

Convolution of A280540 and A280541.

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

Cf. A000005, A038040, A156616, A280540, A280541, A301554, A301555.

a:=series(mul(mul(((1+x^(i*j))/(1-x^(i*j)))^(i*j),j=1..100),i=1..100),x=0,31): seq(coeff(a,x,n),n=0..30); # Paolo P. Lava, Apr 02 2019

nmax = 30; CoefficientList[Series[Product[Product[((1 + x^(i j))/(1 - x^(i j)))^(i j), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(k DivisorSigma[0, k]), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(1 - (-1)^(k/d)) d^2 DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 30}]

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k*tau(k)), where tau(k) = number of divisors of k (A000005).
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (1 - (-1)^(k/d))*d^2*tau(d) ) * x^k/k).
log(a(n)) ~ 3^(2/3) * (7*Zeta(3))^(1/3) * log(n)^(1/3) * n^(2/3) / 2^(4/3). - Vaclav Kotesovec, Sep 03 2018

A320237 G.f.: Product_{k>=1, j>=1} ((1 + x^(kj)) / (1 - x^(kj)))^2.

Original entry on oeis.org

1, 4, 16, 52, 156, 428, 1120, 2772, 6616, 15224, 34032, 74020, 157340, 327244, 667824, 1338828, 2641332, 5133372, 9840432, 18621476, 34818852, 64374564, 117768176, 213306948, 382733816, 680630120, 1200198784, 2099417544, 3644332860, 6280017740, 10746594208

Offset: 0

Vaclav Kotesovec, Oct 08 2018

nonn

Self-convolution of A301554.

Convolution of A320235 and A320236.

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

Cf. A301554, A320235, A320236.

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

Conjecture: log(a(n)) ~ Pi * sqrt(n*log(n)).

A321057 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 - x^k))^sigma_n(k).

Original entry on oeis.org

1, 2, 12, 94, 1522, 48154, 3087600, 377880794, 93356591804, 46415548879976, 44773963087975388, 86770399797767582434, 340765670578000502365102, 2625605734866823121935402410, 40755373130582885082115865730892, 1290109927277547765958474680645604818

Offset: 0

Seiichi Manyama, Oct 26 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..80

Cf. A301554, A301555, A301556, A319647, A321042, A321068.

Table[SeriesCoefficient[Product[((1 + x^k)/(1 - x^k))^DivisorSigma[n, k], {k, 1, n}], {x, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Oct 27 2018 *)

{a(n) = polcoeff(prod(k=1, n, ((1+x^k+x*O(x^n))/(1-x^k+x*O(x^n)))^sigma(k, n)), n)}

A321240 Expansion of Product_{i>=1, j>=1, k>=1, l>=1} (1 + x^(ijkl))/(1 - x^(ijkl)).

Original entry on oeis.org

1, 2, 10, 26, 86, 210, 594, 1394, 3530, 8006, 18842, 41258, 92190, 195714, 419538, 867050, 1797568, 3625758, 7311382, 14431294, 28416514, 55010142, 106101558, 201814518, 382213566, 715473554, 1333083950, 2459265058, 4515151234, 8218572030, 14888270366, 26766878302

Offset: 0

Seiichi Manyama, Nov 01 2018

nonn

Convolution of the sequences A280486 and A280487.

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

Cf. A007426, A015128, A280486, A280487, A301554, A305050.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(   (&*[(&*[(&*[(&*[(1+x^(i*j*k*l))/(1-x^(i*j*k*l)): i in [1..m]]): j in [1..m]]): k in [1..m]]): l in [1..m]]))); // G. C. Greubel, Nov 01 2018

With[{nmax=50}, CoefficientList[Series[Product[(1 + x^(i*j*k*l))/(1 - x^(i*j*k*l)), {i,1,nmax}, {j,1,nmax/i}, {k,1,nmax/i/j}, {l,1,nmax/i/j/k}], {x,0,nmax}], x]] (* G. C. Greubel, Nov 01 2018 *)

m=50; x='x+O('x^m); Vec(prod(k=1,m, ((1+x^k)/(1-x^k))^ sumdiv(k, d, numdiv(k/d)*numdiv(d)))) \\ G. C. Greubel, Nov 01 2018

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A007426(k).

A302238 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^prime(k).

Original entry on oeis.org

1, 4, 14, 46, 136, 382, 1022, 2626, 6530, 15784, 37218, 85842, 194146, 431358, 943038, 2031454, 4316884, 9058662, 18787730, 38542526, 78264298, 157403290, 313712482, 619919350, 1215125262, 2363570168, 4563951858, 8751621598, 16670498062, 31553539214, 59361428202

Offset: 0

Ilya Gutkovskiy, Apr 03 2018

nonn

Convolution of the sequences A030009 and A061152.

N. J. A. Sloane, Transforms

Cf. A000040, A015128, A030009, A061152, A156616, A206622, A206623, A206624, A260916, A261386, A261452, A261519, A261520, A301554, A301555.

nmax = 30; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^Prime[k], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A000040(k).

A302239 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^p(k), where p(k) = number of partitions of k (A000041).

Original entry on oeis.org

1, 2, 6, 16, 40, 96, 226, 512, 1140, 2488, 5336, 11270, 23494, 48356, 98438, 198338, 395846, 783136, 1536800, 2992818, 5786952, 11114950, 21213906, 40247696, 75928804, 142475644, 265985628, 494155176, 913802164, 1682338192, 3084101744, 5630853218, 10240484332, 18553818210

Offset: 0

Ilya Gutkovskiy, Apr 03 2018

nonn

Convolution of the sequences A001970 and A261049.

N. J. A. Sloane, Transforms

Cf. A000041, A001970, A015128, A156616, A206622, A206623, A206624, A260916, A261049, A261386, A261452, A261519, A261520, A301554, A301555, A302237, A302238.

nmax = 33; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^PartitionsP[k], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A000041(k).

cp's OEIS Frontend

A302237 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k*(k+1)/2).

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A308286 Expansion of Product_{i>=1, j>=1} theta_3(x^(i*j)), where theta_3() is the Jacobi theta function.

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A308288 Expansion of Product_{i>=1, j>=1} theta_3(x^(i*j))/theta_4(x^(i*j)), where theta_() is the Jacobi theta function.

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A318579 Expansion of Product_{i>=1, j>=1} ((1 + x^(i*j))/(1 - x^(i*j)))^(i*j).

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Maple

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A320237 G.f.: Product_{k>=1, j>=1} ((1 + x^(k*j)) / (1 - x^(k*j)))^2.

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A321057 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 - x^k))^sigma_n(k).

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A321240 Expansion of Product_{i>=1, j>=1, k>=1, l>=1} (1 + x^(i*j*k*l))/(1 - x^(i*j*k*l)).

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Magma

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A302238 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^prime(k).

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A308288 Expansion of Product_{i>=1, j>=1} theta_3(x^(ij))/theta_4(x^(ij)), where theta_() is the Jacobi theta function.

A318579 Expansion of Product_{i>=1, j>=1} ((1 + x^(ij))/(1 - x^(ij)))^(i*j).

A320237 G.f.: Product_{k>=1, j>=1} ((1 + x^(kj)) / (1 - x^(kj)))^2.

A321240 Expansion of Product_{i>=1, j>=1, k>=1, l>=1} (1 + x^(ijkl))/(1 - x^(ijkl)).