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.

A132975 Expansion of q * psi(-q^9) / psi(-q) in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 12, 15, 20, 26, 32, 39, 50, 63, 76, 92, 114, 140, 168, 201, 244, 295, 350, 415, 496, 591, 696, 818, 967, 1140, 1332, 1554, 1820, 2126, 2468, 2861, 3324, 3855, 4448, 5126, 5916, 6816, 7824, 8970, 10292, 11793, 13471, 15372, 17548
Offset: 1

Views

Author

Michael Somos, Sep 07 2007

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A128129, A128640, A132302, A132972, A132976. Essentially the same as A213267.

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(9*k)) * (1+x^(18*k)) / (1-x^(4*k)),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(9/2)] / EllipticTheta[ 2, Pi/4, q^(1/2)], {q, 0, n}]; (* Michael Somos, Oct 31 2015 *)
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^9 + A) * eta(x^36 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^18 + A)), n))};

Formula

Expansion of eta(q^2) * eta(q^9) * eta(q^36) / (eta(q) * eta(q^4) * eta(q^18)) in powers of q.
Euler transform of period 36 sequence [ 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1 * u2 - (1 + u1 + u2) * (u3 + u6 + 3 * u3 * u6).
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 A132976.
G.f.: x * Product_{k>0} P(3,x^k) * P(9,x^k) * P(12,x^k) * P(36,x^k) where P(n,x) is the n-th cyclotomic polynomial.
3 * a(n) = A132972(n) unless n=0. a(2*n) = A128129(n). a(2*n + 1) = A132302(n). a(3*n) = A128640(n). Convolution inverse of A132976.
a(n) ~ exp(2*Pi*sqrt(n)/3) / (6 * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

A128641 Expansion of (1/3) * (c(q)^2 / c(q^2)) / (b(q^2)^2 / b(q)) in powers of q where b(), c() are cubic AGM theta functions.

Original entry on oeis.org

1, -1, 4, -10, 20, -39, 76, -140, 244, -415, 696, -1140, 1820, -2861, 4448, -6816, 10292, -15372, 22756, -33356, 48408, -69683, 99600, -141312, 199036, -278557, 387608, -536230, 737632, -1009464, 1374888, -1863764, 2514868, -3378948, 4521672, -6027000, 8002676
Offset: 0

Views

Author

Michael Somos, Mar 16 2007

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 + 4*q^2 - 10*q^3 + 20*q^4 - 39*q^5 + 76*q^6 - 140*q^7 + ...

Links

Crossrefs

Cf. A128636, A128640, A164617.

Programs

  • Mathematica
    eta[x_] := x^(1/24)*QPochhammer[x]; A128641[n_] := SeriesCoefficient[ (eta[q]/eta[q^6])*(eta[q^3]/eta[q^2])^5, {q, 0, n}]; Table[A128641[n], {n, 0, 50}] (* G. C. Greubel, Aug 22 2017 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) / eta(x^2 + A))^5 * eta(x + A) / eta(x^6 + A), n))};

Formula

Expansion of (phi(-q^3)^3 / phi(-q)) / (psi(q)^3 / psi(q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (eta(q) / eta(q^6)) * (eta(q^3) / eta(q^2))^5 in powers of q.
Euler transform of period 6 sequence [ -1, 4, -6, 4, -1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u * (1-v) * (8-9*u) + (u-v)^2.
G.f.: Product_{k>0} (1 - x^k) / (1 - x^(6*k)) * ((1 - x^(3*k)) / (1 - x^(2*k)))^5.
A128640(n) = -a(n) unless n = 0. Convolution inverse of A128636.
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 2/3 - (2/9)*sqrt(3) + (2/9)*sqrt(6)*3^(1/4). - Simon Plouffe, Mar 04 2021
a(n) = (-1)^n*A164617(n). - Michael Somos, Apr 24 2023

A128637 Expansion of 3 * (b(q)^2/b(q^2)) / (c(q)^2/c(q^2)) in powers of q where b(), c() are cubic AGM theta functions.

Original entry on oeis.org

1, -8, 24, -24, -40, 144, -120, -192, 600, -456, -688, 2016, -1464, -2096, 5952, -4176, -5800, 15984, -10920, -14816, 39888, -26688, -35488, 93888, -61752, -80824, 210576, -136536, -176320, 453456, -290448, -370688, 942936, -597600, -755024, 1901952, -1194216
Offset: 0

Views

Author

Michael Somos, Mar 16 2007

Keywords

Comments

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

Examples

			G.f. = 1 - 8*q + 24*q^2 - 24*q^3 - 40*q^4 + 144*q^5 - 120*q^6 - 192*q^7 + ...

Links

Crossrefs

Cf. A123633, A128639, A128640.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q] / EllipticTheta[ 4, 0, q^3])^4, {q, 0, n}]; (* Michael Somos, Apr 06 2013 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q^3, q^3] / QPochhammer[ -q, q])^4 /(QPochhammer[ q^3] / QPochhammer[ q])^4, {q, 0, n}]; (* Michael Somos, Apr 06 2013 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x + A) / eta(x^3 + A))^2 * eta(x^6 + A) / eta(x^2 + A))^4, n))};

Formula

Expansion of (phi(-q) / phi(-q^3))^4 in powers of q where phi() is a Ramanujan theta function.
Expansion of ((eta(q) / eta(q^3))^2 * (eta(q^6) / eta(q^2)))^4 in powers of q.
Euler transform of period 6 sequence [ -8, -4, 0, -4, -8, 0, ...].
G.f.: A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u * (1-v) * (9-v) - (u-v)^2.
G.f.: A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u^2*v^2 + 9*u*v - 12*u*v^2 + 30*v^2 - 108*v + 81) * u - v^3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 9 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128640.
G.f.: (Product_{k>0} (1 - x^k + x^(2*k)) / (1 + x^k + x^(2*k)) )^4.
a(n) = -8 * A123633(n) unless n=0. Convolution inverse of A128639.
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 9 + 3*sqrt(3) - 3*sqrt(9+6*sqrt(3)). - Simon Plouffe, Mar 02 2021

A128642 Expansion of (b(q) / b(q^2))^3 in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, -9, 36, -90, 180, -351, 684, -1260, 2196, -3735, 6264, -10260, 16380, -25749, 40032, -61344, 92628, -138348, 204804, -300204, 435672, -627147, 896400, -1271808, 1791324, -2507013, 3488472, -4826070, 6638688, -9085176, 12373992, -16773876, 22633812
Offset: 0

Views

Author

Michael Somos, Mar 16 2007

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 - 9*q + 36*q^2 - 90*q^3 + 180*q^4 - 351*q^5 + 684*q^6 - 1260*q^7 + ...

Links

Crossrefs

Cf. A123633, A128640, A128643.

Programs

  • Mathematica
    eta[q_]:= q^(1/24)*QP[q]; a[n_]:= SeriesCoefficient[((eta[q]/eta[q^2])^3*(eta[q^6]/eta[q^3]))^3, {q, 0, n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Mar 29 2019 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x + A) / eta(x^2 + A))^3 * eta(x^6 + A) / eta(x^3 + A))^3, n))};

Formula

Expansion of (chi(-q)^3 / chi(-q^3))^3 in powers of q where chi() is a Ramanujan theta function.
Expansion of ((eta(q) / eta(q^2))^3 * (eta(q^6) / eta(q^3)))^3 in powers of q.
Euler transform of period 6 sequence [ -9, 0, -6, 0, -9, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u * (1-v) * (8+u) - (u-v)^2.
G.f.: (Product_{k>0} (1 + x^(3*k)) / (1 + x^k)^3)^3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123633.
9*A128640(n) = -a(n) unless n = 0. Convolution inverse of A128643.
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = -2 - 2*3^(1/2) + 2*6^(1/2)*3^(1/4). - Simon Plouffe, Mar 04 2021
Showing 1-4 of 4 results.

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