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 1-4 of 4 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

A233034 Expansion of (f(-x^2) / phi(-x^3))^2 in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 0, -2, 4, -1, -8, 14, -4, -23, 40, -10, -60, 98, -24, -140, 224, -54, -304, 478, -112, -627, 968, -224, -1236, 1884, -432, -2346, 3540, -801, -4320, 6454, -1448, -7742, 11472, -2556, -13548, 19936, -4408, -23226, 33952, -7462, -39080, 56800, -12416, -64660
Offset: 0

Views

Author

Michael Somos, Dec 03 2013

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 - 2*x^2 + 4*x^3 - x^4 - 8*x^5 + 14*x^6 - 4*x^7 - 23*x^8 + 40*x^9 + ...
G.f. = q - 2*q^13 + 4*q^19 - q^25 - 8*q^31 + 14*q^37 - 4*q^43 - 23*q^49 + ...

Links

Crossrefs

Cf. A058531, A062242, A062244, A092848, A093073, A128111, A128143, A128144, A128145, A132976, A139032, A143840, A164268, A164612, A164615, A182032, A182033, A182034, A182035, A182056, A182057, A193261.

Programs

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

Formula

Expansion of q^(-2/3) * b(q^2) * c(q^2) / (3 * f(-q^3)^4) in powers of q where b(), c() are cubic AGM theta functions.
Expansion of q^(-1/6) * (eta(q^2) * eta(q^6) / eta(q^3)^2)^2 in powers of q.
Euler transform of period 6 sequence [ 0, -2, 4, -2, 0, 0, ...].
G.f.: Product_{k>0} ( (1 - x^(2*k)) * (1 - x^(6*k)) / (1 - x^(3*k))^2 )^2.
a(n) = A092848(2*n) = A128111(2*n) = A182057(4*n) = A062242(4*n + 1) = A182056(4*n + 1) = A139032(6*n + 1) = A164615(6*n + 1) = A182033(6*n + 1) = A058531(12*n + 2) = A093073(12*n + 2) = A128143(12*n + 2) = A128145(12*n + 2) = A143840(12*n + 2) = A182032(12*n + 2) = A193261(12*n + 2).
-a(n) = A062244(4*n + 1) = A182034(6*n + 1) = A182035(6*n + 1) = A128144(12*n + 2) = A132976(12*n + 3) = A164268(12*n + 2) = A164612(12*n + 3) = A182035(12*n + 2).

A258111 Expansion of b(q^3) * b(q^12) / (b(-q) * b(q^6)) in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, -3, 9, -24, 57, -126, 264, -528, 1017, -1896, 3438, -6084, 10536, -17898, 29880, -49104, 79545, -127170, 200856, -313692, 484830, -742080, 1125540, -1692648, 2525160, -3738765, 5496246, -8025432, 11643576, -16790310, 24072048, -34321560, 48677625
Offset: 0

Views

Author

Michael Somos, May 20 2015

Keywords

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = 1 - 3*q + 9*q^2 - 24*q^3 + 57*q^4 - 126*q^5 + 264*q^6 - 528*q^7 + ...

Links

Crossrefs

Cf. A164615, A258108.

Programs

  • Mathematica
    QP = QPochhammer; A258111[n_] := SeriesCoefficient[(QP[q]^3*QP[q^3]^2 *QP[q^4]^3*QP[q^12]^2*QP[q^18])/(QP[q^2]^9*QP[q^9]*QP[q^36]), {q, 0, n}]; Table[A258111[n], {n, 0, 50}] (* G. C. Greubel, Oct 18 2017 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^3 + A)^2 * eta(x^4 + A)^3 * eta(x^12 + A)^2 * eta(x^18 + A) / (eta(x^2 + A)^9 * eta(x^9 + A) * eta(x^36 + A)), n))};

Formula

Expansion of eta(q)^3 * eta(q^3)^2 * eta(q^4)^3 * eta(q^12)^2 * eta(q^18) / (eta(q^2)^9 * eta(q^9) * eta(q^36)) in powers of q.
Euler transform of period 36 sequence [ -3, 6, -5, 3, -3, 4, -3, 3, -4, 6, -3, -1, -3, 6, -5, 3, -3, 4, -3, 3, -5, 6, -3, -1, -3, 6, -4, 3, -3, 4, -3, 3, -5, 6, -3, 0, ...]. - Corrected by Sean A. Irvine, Mar 06 2020
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. for A164615.
Convolution inverse of A258108.
a(n) ~ (-1)^n * exp(4*Pi*sqrt(n)/3) / (2^(5/2) * 3^(3/2) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018
Showing 1-4 of 4 results.

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