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 1-7 of 7 results.

A320970 Expansion of Product_{k>0} theta_4(q^k)/theta_3(q^k), where theta_3() and theta_4() are the Jacobi theta functions.

Original entry on oeis.org

1, -4, 4, -4, 20, -28, 20, -52, 84, -104, 156, -180, 308, -460, 468, -684, 1028, -1308, 1592, -2084, 2940, -3668, 4564, -5716, 7556, -9912, 11484, -14616, 19252, -23548, 28316, -35188, 44724, -54532, 65996, -79948, 99784, -122796, 143972, -175372, 216524, -259996, 308004, -371140
Offset: 0

Views

Author

Seiichi Manyama, Oct 25 2018

Keywords

Links

Crossrefs

Cf. A000122, A002448, A320068, A320908, A320967.

Programs

  • Mathematica
    With[{nmax=80}, CoefficientList[Series[Product[EllipticTheta[4, 0, q^k]/EllipticTheta[3, 0, q^k], {k, 1, nmax+2}], {q, 0, nmax}], q]] (* G. C. Greubel, Oct 29 2018 *)
  • PARI
    m=80; q='q+O('q^m); Vec(1/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

Formula

Expansion of Product_{k>0} (eta(q^k)^4*eta(q^(4*k))^2) / eta(q^(2*k))^6.
a(n) ~ (-1)^n * exp(Pi*sqrt(log(2)*n)) * (log(2))^(1/4) / (4*n^(3/4)). - Vaclav Kotesovec, Oct 26 2018

A320971 Expansion of Product_{k>=1} ((1 - x^k)/(1 + x^k))^(sigma(k)).

Original entry on oeis.org

1, -2, -4, 2, 10, 22, -4, -26, -68, -104, -12, 110, 378, 486, 448, -66, -1130, -2242, -3044, -2474, -322, 5106, 11064, 16954, 17896, 10440, -8032, -40132, -74578, -105754, -108564, -66534, 42672, 209858, 421352, 611946, 690204, 553534, 82112, -735082, -1892200
Offset: 0

Views

Author

Seiichi Manyama, Oct 25 2018

Keywords

Links

Crossrefs

Convolution inverse of A301555.
Product_{k>=1} ((1 - x^k)/(1 + x^k))^(sigma_b(k)): A320908 (b=0), this sequence (b=1), A320972 (b=2).
Cf. A000203, A288385, A288421.

Programs

  • Magma
    m:=80; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(  (&*[((1-q^k)/(1+q^k))^DivisorSigma(1,k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 29 2018
  • Mathematica
    With[{nmax=80}, CoefficientList[Series[Product[((1-q^k)/(1+q^k) )^DivisorSigma[1,k], {k, 1, nmax+2}], {q, 0, nmax}], q]] (* G. C. Greubel, Oct 29 2018 *)
  • PARI
    N=99; x='x+O('x^N); Vec(prod(k=1, N, ((1-x^k)/(1+x^k))^sigma(k)))

A320972 Expansion of Product_{k>=1} ((1 - x^k)/(1 + x^k))^(sigma_2(k)).

Original entry on oeis.org

1, -2, -8, -2, 30, 110, 92, -182, -976, -2064, -1488, 3714, 17618, 35814, 37680, -25278, -216910, -541538, -819268, -480334, 1441634, 5924858, 12518720, 16883366, 7972200, -32275008, -120780700, -250726492, -349220282, -229745138, 424373412, 1958370998, 4418456156
Offset: 0

Views

Author

Seiichi Manyama, Oct 25 2018

Keywords

Links

Crossrefs

Convolution inverse of A301556.
Product_{k>=1} ((1 - x^k)/(1 + x^k))^(sigma_b(k)): A320908 (b=0), A320971 (b=1), this sequence (b=2).
Cf. A001157, A288389, A288422.

Programs

  • PARI
    N=99; x='x+O('x^N); Vec(prod(k=1, N, ((1-x^k)/(1+x^k))^sigma(k, 2)))

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.

A321068 a(n) = [x^n] Product_{k>=1} ((1 - x^k)/(1 + x^k))^sigma_n(k).

Original entry on oeis.org

1, -2, -8, -22, 294, 24982, 1372372, 10145326, -38651841784, -21995644478504, -5088041946350856, 29713279339187796814, 155715351422115081062330, 370606511915720675179342334, -12360092915168107023209454901320
Offset: 0

Views

Author

Seiichi Manyama, Oct 26 2018

Keywords

Links

Crossrefs

Cf. A320908, A320971, A320972, A321042, A321057.

Programs

  • Mathematica
    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 *)
  • PARI
    {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)}

A321302 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, -6, 6, 14, 30, -14, -98, -86, -150, 282, 486, 502, 670, -1118, -1226, -4396, -3814, 1326, 3834, 20354, 16330, 18334, -6606, -45658, -60762, -121770, -60122, -22750, 160314, 303638, 435450, 542336, 162782, -45830, -1090994, -1576378, -2608146, -2408142, -988202, 479834
Offset: 0

Views

Author

Seiichi Manyama, Nov 03 2018

Keywords

Links

Crossrefs

Convolution inverse of A321240.
Cf. A007426, A320908, A321241.

Programs

  • PARI
    \\ here b(n) is A007426.
b(n)={vecprod(apply(e->binomial(e+3, 3), factor(n)[,2]))}
seq(n)={Vec(prod(k=1, n, ((1 - x^k)/(1 + x^k) + O(x*x^n))^b(k)))} \\ Andrew Howroyd, Nov 06 2018

Formula

G.f.: Product_{k>=1} ((1 - x^k)/(1 + x^k))^A007426(k).
Showing 1-7 of 7 results.

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

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