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.

A164616 Expansion of c(-q) * c(-q^3) / c(q^2)^2 in powers of q where c() is a cubic AGM theta function.

Original entry on oeis.org

1, -1, 0, 1, -2, 0, 4, -5, 0, 10, -12, 0, 20, -26, 0, 39, -50, 0, 76, -92, 0, 140, -168, 0, 244, -295, 0, 415, -496, 0, 696, -818, 0, 1140, -1332, 0, 1820, -2126, 0, 2861, -3324, 0, 4448, -5126, 0, 6816, -7824, 0, 10292, -11793, 0, 15372, -17548, 0, 22756
Offset: 0

Views

Author

Michael Somos, Aug 17 2009

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).

Examples

			G.f. = 1 - q + q^3 - 2*q^4 + 4*q^6 - 5*q^7 + 10*q^9 - 12*q^10 + 20*q^12 + ...

Links

Crossrefs

Cf. A132977, A164615, A164617, A258100, A258108.

Programs

  • Mathematica
    eta[x_] := QPochhammer[x]; A164616[n_] := SeriesCoefficient[eta[q]* eta[q^4]*eta[q^18]^9/(eta[q^2]*eta[q^3]^2*eta[q^9]^3*eta[q^12]^2* eta[q^36]^3), {q, 0, n}]; Table[A164616[n], {n,0,50}] (* G. C. Greubel, Aug 10 2017 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^18 + A)^9 / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^9 + A)^3 * eta(x^12 + A)^2 * eta(x^36 + A)^3), n))};

Formula

Expansion of (psi(-q) * f(q^9)^3) / (chi(q^3) * psi(-q^3)^2)^2 in powers of q where psi(), chi(), f() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^4) * eta(q^18)^9 / (eta(q^2) * eta(q^3)^2 * eta(q^9)^3 * eta(q^12)^2 * eta(q^36)^3) in powers of q.
Euler transform of period 36 sequence [ -1, 0, 1, -1, -1, 2, -1, -1, 4, 0, -1, 3, -1, 0, 1, -1, -1, -4, -1, -1, 1, 0, -1, 3, -1, 0, 4, -1, -1, 2, -1, -1, 1, 0, -1, 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 A258108. - Michael Somos, May 20 2015
a(3*n + 2) = 0. a(3*n) = A164617(n). a(3*n + 1) = -A132977(n).
Convolution inverse of A164615.
a(n) = (-1)^n * A258100(n). - Michael Somos, May 20 2015

A182034 Expansion of c(q^2)^2 / (c(q) * c(q^3)) in powers of q where c() is a cubic AGM theta function.

Original entry on oeis.org

1, -1, 1, 0, 1, -2, 0, 0, 1, 0, -2, 4, 0, 2, -8, 0, 1, 2, 0, -4, 14, 0, 4, -24, 0, 1, 6, 0, -8, 38, 0, 8, -63, 0, 2, 16, 0, -14, 92, 0, 14, -150, 0, 4, 36, 0, -24, 208, 0, 23, -329, 0, 6, 78, 0, -40, 440, 0, 38, -684, 0, 10, 160, 0, -63, 884, 0, 60, -1358, 0
Offset: 0

Views

Author

Michael Somos, Apr 07 2012

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).

Examples

			G.f. = 1 - q + q^2 + q^4 - 2*q^5 + q^8 - 2*q^10 + 4*q^11 + 2*q^13 - 8*q^14 + ...

Links

Crossrefs

Cf. A092848, A164615, A216046, A258100.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (2 q^(1/8) QPochhammer[ q^6]^6) / (QPochhammer[ q^9]^3 EllipticTheta[ 2, 0, q^(1/2)] QPochhammer[ q^3]^2), {q, 0, n}]; (* Michael Somos, May 20 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3] EllipticTheta[ 2, 0, q^(3/2)]^3) / (4 q QPochhammer[ q^9]^3 EllipticTheta[ 2, 0, q^(1/2)]), {q, 0, n}]; (* Michael Somos, May 20 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^6 / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^9 + A)^3), n))};

Formula

Expansion of (chi(-q^3)^2 * psi(q^3)^4) / (psi(q) * f(-q^9)^3) in powers of q where psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, May 20 2015
Expansion of eta(q) * eta(q^6)^6 / (eta(q^2)^2 * eta(q^3)^2 * eta(q^9)^3) in powers of q.
Euler transform of period 18 sequence [ -1, 1, 1, 1, -1, -3, -1, 1, 4, 1, -1, -3, -1, 1, 1, 1, -1, 0, ...].
a(n) = (-1)^n * A164615(n). a(3*n) = 0 unless n=0. a(3*n + 1) = - A092848(n). a(3*n + 2) = A216046(n).
Convolution inverse is A258100. - Michael Somos, May 20 2015

A258099 Expansion of ( psi(x^3) * phi(-x^3) / (psi(x) * f(-x^2)) )^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 5, -12, 26, -50, 92, -168, 295, -496, 818, -1332, 2126, -3324, 5126, -7824, 11793, -17548, 25857, -37788, 54734, -78578, 111968, -158496, 222842, -311224, 432095, -596676, 819504, -1119624, 1522282, -2060448, 2776514, -3725294, 4978142, -6626988
Offset: 0

Views

Author

Michael Somos, May 20 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 + 26*x^4 - 50*x^5 + 92*x^6 - 168*x^7 + ...
G.f. = q - 2*q^4 + 5*q^7 - 12*q^10 + 26*q^13 - 50*q^16 + 92*q^19 + ...

Links

Crossrefs

Cf. A132977, A258100.

Programs

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

Formula

Expansion of q^(-1/3) * (eta(q) * eta(q^3) * eta(q^6) / eta(q^2)^3)^2 in powers of q.
Euler transform of period 6 sequence [ -2, 4, -4, 4, -2, 0, ...].
G.f.: Product_{k>0} (1 - x^k + x^(2*k))^2 * (1 + x^k + x^(2*k))^4 / (1 + x^k)^4.
a(n) = A258100(3*n + 1).
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.