cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A321241 Expansion of Product_{i>=1, j>=1, k>=1} (1 - x^(i*j*k))/(1 + x^(i*j*k)).

Original entry on oeis.org

1, -2, -4, 4, 8, 16, -12, -28, -28, -56, 64, 68, 152, 144, -20, -72, -678, -508, -424, 92, 824, 1512, 2204, 1036, 936, -1900, -2936, -6444, -5656, -4384, -4808, 6540, 10080, 21256, 20296, 24424, 13520, -7856, -28836, -55744, -72240, -92960, -48424, -40772, 36168, 106464, 199996
Offset: 0

Views

Author

Seiichi Manyama, Nov 01 2018

Keywords

Comments

Convolution inverse of A305050.

Links

Crossrefs

Cf. A000005, A002448, A007425, A305050, A320908.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(   (&*[(&*[(&*[(1 - x^(i*j*k))/(1 + x^(i*j*k)): i in [1..m]]): j in [1..m]]): k in [1..m]]))); // G. C. Greubel, Nov 01 2018
  • Mathematica
    With[{nmax=50}, 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]] (* G. C. Greubel, Nov 01 2018 *)
  • PARI
    m=50; x='x+O('x^m); Vec(prod(k=1,m, ((1-x^k)/(1+x^k))^sumdiv(k, x, sumdiv(x, y, 1 )))) \\ G. C. Greubel, Nov 01 2018

Formula

G.f.: Product_{k>=1} ((1 - x^k)/(1 + x^k))^A007425(k).
G.f.: Product_{k>=1} theta_4(x^k)^tau(k), where theta_4() is the Jacobi theta function and tau() is the number of divisors. - Ilya Gutkovskiy, May 18 2019

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

Views

Author

Ilya Gutkovskiy, May 18 2019

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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]

Formula

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^(i*j))/theta_4(x^(i*j)), 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

Views

Author

Ilya Gutkovskiy, May 18 2019

Keywords

Comments

Convolution of the sequences A305050 and A308286.

Links

Crossrefs

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

Programs

  • Mathematica
    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]

Formula

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.

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

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

Views

Author

Seiichi Manyama, Nov 01 2018

Keywords

Comments

Convolution of the sequences A280486 and A280487.

Links

Crossrefs

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

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • PARI
    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

Formula

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

A318764 Expansion of Product_{i>=1, j>=1, k>=1} ((1 + x^(i*j*k))/(1 - x^(i*j*k)))^(i*j*k).

Original entry on oeis.org

1, 2, 14, 44, 182, 548, 1932, 5632, 17654, 49872, 145020, 395256, 1090044, 2876424, 7606024, 19503312, 49850790, 124543772, 309436980, 755268832, 1831194724, 4376807896, 10387118328, 24359228520, 56720659372, 130737105940, 299256890672, 678941040784
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 03 2018

Keywords

Comments

Convolution of A318413 and A318414.

Links

Crossrefs

Cf. A318413, A318414.
Cf. A156616, A318579.
Cf. A015128, A305050.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[Product[Product[((1 + x^(i*j*k))/(1 - x^(i*j*k)))^(i*j*k), {i, 1, nmax/j/k}], {j, 1, nmax/k}], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(k*Sum[DivisorSigma[0, d], {d, Divisors[k]}]), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

Conjecture: log(a(n)) ~ (21*Zeta(3))^(1/3) * log(n)^(2/3) * n^(2/3) / 2^(5/3).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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