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 1-3 of 3 results.

A320968 Expansion of (Product_{k>0} theta_3(q^k)/theta_4(q^k))^(1/2), where theta_3() and theta_4() are the Jacobi theta functions.

Original entry on oeis.org

1, 2, 4, 10, 18, 34, 64, 110, 188, 320, 524, 846, 1358, 2130, 3308, 5102, 7750, 11674, 17468, 25862, 38022, 55558, 80532, 116034, 166284, 236784, 335416, 472868, 663146, 925762, 1286920, 1780962, 2454792, 3370806, 4610656, 6284090, 8535868, 11554834, 15591564
Offset: 0

Views

Author

Seiichi Manyama, Oct 25 2018

Keywords

Links

Crossrefs

Cf. A000122, A002448, A080054 ((theta_3(q^k)/theta_4(q^k))^(1/2)), A320098, A320967, A320992.

Programs

  • Mathematica
    CoefficientList[Series[1/Product[EllipticTheta[4, 0, q^(2*k - 1)], {k, 1, 50}], {q, 0, 80}], q] (* G. C. Greubel, Oct 29 2018 *)
  • PARI
    q='q+O('q^80); Vec(prod(k=1,50, eta(q^(2*k))^3/(eta(q^k)^2* eta(q^(4*k))) )) \\ G. C. Greubel, Oct 29 2018

Formula

a(n) = (-1)^n * A320098(n).
Expansion of Product_{k>0} eta(q^(2*k))^3 / (eta(q^k)^2*eta(q^(4*k))).
Expansion of Product_{k>0} 1/theta_4(q^(2*k-1)).

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

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.
Showing 1-3 of 3 results.

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