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.

A139135 Expansion of psi(-q^3) / f(q) where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -1, 2, -4, 6, -9, 14, -20, 29, -42, 58, -80, 110, -148, 198, -264, 347, -454, 592, -764, 982, -1257, 1598, -2024, 2554, -3206, 4010, -5000, 6208, -7684, 9484, -11664, 14306, -17501, 21346, -25972, 31526, -38170, 46112, -55588, 66861, -80258, 96154, -114968, 137212
Offset: 0

Views

Author

Michael Somos, Apr 10 2008

Keywords

Comments

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Examples

			q - q^4 + 2*q^7 - 4*q^10 + 6*q^13 - 9*q^16 + 14*q^19 - 20*q^22 + 29*q^25 + ...

Links

Crossrefs

A139136(3*n + 1) = - a(n). A139137(3*n + 1) = 2 * a(n).
Apart from signs, same as A097197.

Programs

  • Mathematica
    A139135[n_] := SeriesCoefficient[(QPochhammer[q]* QPochhammer[q^3]*QPochhammer[q^4]*QPochhammer[q^12])/(QPochhammer[q^2]^3 *QPochhammer[q^6]), {q, 0, n}]; Table[A139135[n], {n, 0, 50}] (* G. C. Greubel, Oct 05 2017 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x^2 + A)^3 * eta(x^6 + A)), n))}

Formula

Expansion of q^(-1/3) * eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q^2)^3 * eta(q^6)) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (108 t)) = 3^(-1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A139136.
a(n) ~ (-1)^n * exp(Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A262930 Expansion of (psi(-q) / f(q^3))^2 in powers of q where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 1, -4, 6, -2, 12, -16, 5, -28, 36, -12, 60, -76, 24, -120, 150, -46, 228, -280, 86, -416, 504, -152, 732, -878, 262, -1252, 1488, -442, 2088, -2464, 725, -3408, 3996, -1168, 5460, -6364, 1852, -8600, 9972, -2886, 13344, -15400, 4436, -20424, 23472
Offset: 0

Views

Author

Michael Somos, Oct 04 2015

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 - 2*x + x^2 - 4*x^3 + 6*x^4 - 2*x^5 + 12*x^6 - 16*x^7 + 5*x^8 + ...

Links

Crossrefs

Cf. A139136, A261325, A261328, A261369, A263528, A263538.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (1/2) q^(-1/4) (EllipticTheta[ 2, Pi/4, q^(1/2)] / QPochhammer[ -q^3])^2, {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / ( eta(x^2 + A) * eta(x^6 + A)^3 ))^2, n))};

Formula

Expansion of ( eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / ( eta(q^2) * eta(q^6)^3 ))^2 in powers of q.
Euler transform of period 12 sequence [ -2, 0, -4, -2, -2, 4, -2, -2, -4, 0, -2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A261369.
a(3*n) = A261320(n). a(3*n + 1) = -2 * A261325(n). a(3*n + 2) = A261369(n).
Convolution square of A139136.
a(2*n) = A263538(n). a(2*n + 1) = -2 * A263528(n).

A132002 Expansion of phi(q^3) / phi(q) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 4, -6, 10, -16, 24, -36, 52, -74, 104, -144, 198, -268, 360, -480, 634, -832, 1084, -1404, 1808, -2316, 2952, -3744, 4728, -5946, 7448, -9294, 11556, -14320, 17688, -21780, 26740, -32736, 39968, -48672, 59122, -71644, 86616, -104484, 125768, -151072
Offset: 0

Views

Author

Michael Somos, Aug 06 2007

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 - 2*x + 4*x^2 - 6*x^3 + 10*x^4 - 16*x^5 + 24*x^6 - 36*x^7 + 52*x^8 + ...

Links

Crossrefs

Cf. A098151, A139007, A139136, A139137, A261320.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3] / EllipticTheta[ 3, 0, q], {q, 0, n}]; (* Michael Somos, Apr 04 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q, -q^3] QPochhammer[ -q^2, -q^3] / (QPochhammer[ -q, -q^3] QPochhammer[ q^2, -q^3]), {q, 0, n}]; (* Michael Somos, Nov 02 2015 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( sum(k=1, sqrtint(n\3), 2*x^(3*k^2), 1 + x * O(x^n)) / sum(k=1, sqrtint(n), 2*x^k^2, 1 + x * O(x^n)), n))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^5 / (eta(x^2 + A)^5 * eta(x^3 + A)^2 * eta(x^12 + A)^2), n))};

Formula

Expansion of eta(q)^2 * eta(q^4)^2 * eta(q^6)^5 / ( eta(q^2)^5 * eta(q^3)^2 * eta(q^12)^2) in powers of q.
Euler transform of period 12 sequence [ -2, 3, 0, 1, -2, 0, -2, 1, 0, 3, -2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v+u) * (v-u) + (1 - u*v) * (1 - 3*u*v).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^3 - v + 3*u*v^2 * (1 - u*v).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139007. - Michael Somos, Apr 04 2015
G.f.: (Sum_{k in Z} x^(3*k^2)) / (Sum_{k in Z} x^k^2).
G.f.: Product_{k>0} (1 + (-x)^k + x^(2*k)) / (1 - (-x)^k + x^(2*k)). - Michael Somos, Apr 04 2015
a(n) = (-1)^n * A098151(n).
Convolution inverse of A139137. Convolution square is A261320. - Michael Somos, Aug 14 2015
Expansion of f(-q, q^2) / f(q, -q^2) in powers of q where f(, ) is Ramanujan's general theta function. - Michael Somos, Nov 02 2015
a(n) = A139136(3*n) = A139137(3*n). - Michael Somos, Nov 02 2015
a(n) ~ (-1)^n * exp(Pi*sqrt(2*n/3)) / (2^(5/4) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 16 2017
Showing 1-3 of 3 results.

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