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.

A261325 Expansion of f(x^3, x^3) * f(x, x^5) / f(x, x)^2 in powers of x where f(,) is Ramanujan's general theta function.

Original entry on oeis.org

1, -3, 8, -18, 38, -75, 140, -252, 439, -744, 1232, -1998, 3182, -4986, 7700, -11736, 17673, -26322, 38808, -56682, 82070, -117867, 167996, -237744, 334202, -466836, 648224, -895014, 1229148, -1679436, 2283568, -3090672, 4164578, -5587941, 7467464, -9940482
Offset: 0

Views

Author

Michael Somos, Aug 14 2015

Keywords

Comments

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

Examples

			G.f. = 1 - 3*x + 8*x^2 - 18*x^3 + 38*x^4 - 75*x^5 + 140*x^6 - 252*x^7 + ...
G.f. = q - 3*q^4 + 8*q^7 - 18*q^10 + 38*q^13 - 75*q^16 + 140*q^19 + ...

Links

Crossrefs

Cf. A187153, A213265, A233670, A233672, A233673, A233698, A260215.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ -x^3] QPochhammer[ x^6] / QPochhammer[ -x]^3, {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^4 + A)^3 *  eta(x^6 + A)^4 / (eta(x^2 + A)^8 * eta(x^3 + A) * eta(x^12 + A)), n))};

Formula

Expansion of f(-x^2) * f(x^3) * f(-x^6) / f(x)^3 in powers of x where f() is a Ramanujan theta function.
Expansion of q^(-1/3) * eta(q)^3 * eta(q^4)^3 * eta(q^6)^4 / (eta(q^2)^8 * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -3, 5, -2, 2, -3, 2, -3, 2, -2, 5, -3, 0, ...].
a(n) = A187153(3*n + 1) = A213265(3*n + 1) = A233670(3*n + 1) = A233672(3*n + 1).
2 * a(n) = A233673(3*n + 1) = - A260215(3*n + 1). a(2*n + 1) = -3 * A233698(n).
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (4*3^(5/4)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A187153 Expansion of q * (psi(q) / psi(q^2)) / (psi(q^3) / psi(q^6))^3 in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, -1, -3, -2, 3, 8, 5, -7, -18, -12, 15, 38, 24, -30, -75, -46, 57, 140, 86, -104, -252, -152, 183, 439, 262, -313, -744, -442, 522, 1232, 725, -852, -1998, -1168, 1365, 3182, 1852, -2150, -4986, -2886, 3336, 7700, 4436, -5106, -11736, -6736, 7719
Offset: 1

Views

Author

Michael Somos, Mar 06 2011

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
The Alaca, et. al. paper in equation (1.8) on page 178 uses the notation p(q) := (phi^2(q) - phi^2(q^3))/(2*phi^2(q^3)), where phi(q) is a Ramanujan theta function. This p(q) is twice the g.f. of this sequence. - Michael Somos, Mar 30 2021

Examples

			G.f. = q + q^2 - q^3 - 3*q^4 - 2*q^5 + 3*q^6 + 8*q^7 + 5*q^8 - 7*q^9 - 18*q^10 + ...

Links

Crossrefs

Cf. A115977, A123649, A187143, A213265.

Programs

  • Mathematica
    QP = QPochhammer; s = (QP[q^2]*QP[q^3]*QP[q^12]^2)^3/(QP[q]*QP[q^4]^2* QP[q^6]^9) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
QP = QPochhammer; Rest[Table[SeriesCoefficient[q*(QP[-q, q^2]*QP[-q^6, q^6]^3)/(QP[-q^2, q^2]*QP[-q^3, q^6]^3), {q, 0, n}], {n, 0, 50}]] (* G. C. Greubel, Dec 04 2017 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ x^2, x^4]^2 QPochhammer[x^3, x^6]^3 QPochhammer[ -x^6, x^6]^6, {x, 0, n}]; (* Michael Somos, Oct 15 2018 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[3, 0, x]^2 / EllipticTheta[3, 0, x^3]^2 - 1)/4, {x, 0, n}]; (* Michael Somos, Mar 30 2021 *)
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)^2)^3 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)^9), n))};

Formula

Expansion of (eta(q^2) * eta(q^3) * eta(q^12)^2)^3 / (eta(q) * eta(q^4)^2 * eta(q^6)^9) in powers of q.
Euler transform of period 12 sequence [1, -2, -2, 0, 1, 4, 1, 0, -2, -2, 1, 0, ...].
Expansion of c(q) * c(q^4)^2 / c(q^2)^3 in powers of q where c() is a cubic AGM theta function.
If p = 2 * A(q), then B(q) = p * ((2 + p) / (1 + 2*p))^3 and B(q^3) = p^3 * ((2 + p) / (1 + 2*p)) where B() is the g.f. for A115977. - Michael Somos, Feb 27 2012
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u * (1 + 2*v))^2 - v * (1 + 2*u).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^2 * (1 + 4*v) * (3*v + u*(1 + 4*v)) - v * (1 + v) * (3*u + 1 + v).
Convolution inverse of A187143.
Expansion of (phi^2(q) - phi^2(q^3))/(4*phi^2(q^3)), where phi(q) is a Ramanujan theta function. - Michael Somos, Mar 30 2021

A261369 Expansion of (psi(-x^3) / f(x))^2 in powers of x where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 5, -12, 24, -46, 86, -152, 262, -442, 725, -1168, 1852, -2886, 4436, -6736, 10103, -14994, 22040, -32092, 46336, -66380, 94378, -133256, 186926, -260576, 361126, -497716, 682340, -930774, 1263624, -1707672, 2297737, -3078850, 4109022, -5462924, 7236280
Offset: 0

Views

Author

Michael Somos, Aug 16 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 + 5*x^2 - 12*x^3 + 24*x^4 - 46*x^5 + 86*x^6 - 152*x^7 + ...
G.f. = q^2 - 2*q^5 + 5*q^8 - 12*q^11 + 24*q^14 - 46*q^17 + 86*q^20 + ...

Links

Crossrefs

Cf. A139135, A187153, A213265, A262930.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x^(3/2)]^2 / (2 x^(3/4) QPochhammer[ -x]^2), {x, 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)^3 * eta(x^6 + A)))^2, n))};

Formula

Expansion of q^(-2/3) * (eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q^2)^3 * eta(q^6)))^2 in powers of q.
Euler transform of period 12 sequence [ -2, 4, -4, 2, -2, 4, -2, 2, -4, 4, -2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A262930. - Michael Somos, Nov 07 2015
a(n) = A187153(3*n + 2) = A213265(3*n + 2) = A262930(3*n + 2). - Michael Somos, Nov 07 2015
Convolution square of A139135. - Michael Somos, Nov 07 2015
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (8*3^(5/4)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A259033 Expansion of psi(x^3)^2 * f(-x^2)^4 / f(-x)^6 in powers of where psi(), f() are Ramanujan theta function.

Original entry on oeis.org

1, 6, 23, 76, 221, 584, 1443, 3368, 7497, 16046, 33190, 66628, 130288, 248858, 465387, 853836, 1539425, 2731462, 4775703, 8236856, 14027754, 23609794, 39301171, 64747876, 105638153, 170778512, 273704800, 435079524, 686237877, 1074405242, 1670333294, 2579418528
Offset: 0

Views

Author

Michael Somos, Nov 07 2015

Keywords

Comments

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

Examples

			G.f. = 1 + 6*x + 23*x^2 + 76*x^3 + 221*x^4 + 584*x^5 + 1443*x^6 + 3368*x^7 + ...
G.f. = q^5 + 6*q^11 + 23*q^17 + 76*q^23 + 221*q^29 + 584*q^35 + 1443*q^41 + ...

Links

Crossrefs

Cf. A213265, A261369, A262930, A263528.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (1/4) x^(-3/4) EllipticTheta[ 2, 0, x^(3/2)]^2 QPochhammer[ x^2]^4 / QPochhammer[ x]^6, {x, 0, n}];
nmax = 40; CoefficientList[Series[Product[((1 + x^k)^2 * (1 + x^(3*k))^2 * (1 - x^(3*k)) / (1 - x^k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 16 2017 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^6 + A)^2 / (eta(x + A)^3 * eta(x^3 + A)))^2, n))};

Formula

Expansion of q^(-5/6) * (eta(q^2)^2 * eta(q^6)^2 / (eta(q)^3 * eta(q^3)))^2 in powers of q.
Euler transform of period 6 sequence [ 6, 2, 8, 2, 6, 0, ...].
a(n) = A263528(3*n + 2). -2 * a(n) = A261369(2*n + 1) = A213265(6*n + 5) = A262930(6*n + 5).
a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(19/4) * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A261321 Expansion of (phi(q) / phi(q^3))^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 4, 4, -4, -12, -8, 12, 32, 20, -28, -72, -48, 60, 152, 96, -120, -300, -184, 228, 560, 344, -416, -1008, -608, 732, 1756, 1048, -1252, -2976, -1768, 2088, 4928, 2900, -3408, -7992, -4672, 5460, 12728, 7408, -8600, -19944, -11544, 13344, 30800, 17744, -20424
Offset: 0

Views

Author

Michael Somos, Aug 14 2015

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
The generating function is associated with a modular equation of degree 3 and is the multiplier denoted by "m". - Michael Somos, Nov 01 2017

Examples

			G.f. = 1 + 4*x + 4*x^2 - 4*x^3 - 12*x^4 - 8*x^5 + 12*x^6 + 32*x^7 + ...

References

  • B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 230 Entry 5(iii), g.f. denoted by multiplier m.

Links

Crossrefs

Cf. A123633, A128636, A139137, A187153, A213265, A217771, A228447, A261320.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^3])^2, {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^10 * eta(x^3 + A)^4 * eta(x^12 + A)^4 / (eta(x + A)^4 * eta(x^4 + A)^4 * eta(x^6 + A)^10), n))};

Formula

Expansion of eta(q^2)^10 * eta(q^3)^4 * eta(q^12)^4 / (eta(q)^4 * eta(q^4)^4 * eta(q^6)^10) in powers of q.
G.f.: (Sum_{k in Z} x^k^2) / (Sum_{k in Z} x^(3*k^2))^2.
a(n) = -(1)^n * A217771(n). a(n) = 4 * A187153(n) = 4 * A213265(n) unless n=0.
a(2*n) = 4 * A123633(n) = 4 * A128636(n) unless n=0. a(3*n) = -4 * A228447(n) unless n=0.
Convolution inverse is A261320. Convolution square of A139137.
Showing 1-5 of 5 results.

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