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.

A164617 Expansion of (phi^3(q^3) / phi(q)) * (psi(-q^3) / psi^3(-q)) in powers of q where phi(), psi() are Ramanujan 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, Aug 17 2009

Keywords

Comments

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

Examples

			G.f. = 1 + q + 4*q^2 + 10*q^3 + 20*q^4 + 39*q^5 + 76*q^6 + 140*q^7 + 244*q^8 + ...

Links

Crossrefs

Cf. A113973, A128641, A132974, A164616.

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Product[(1-x^(6*k))^14 / ((1-x^k) * (1-x^(2*k))^2 * (1-x^(3*k))^5 * (1-x^(4*k)) * (1-x^(12*k))^5),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^14 / (eta(x + A) * eta(x^2 + A)^2 * eta(x^3 + A)^5 * eta(x^4 + A) * eta(x^12 + A)^5), n))};

Formula

Expansion of eta(q^6)^14 / (eta(q) * eta(q^2)^2 * eta(q^3)^5 * eta(q^4) * eta(q^12)^5) in powers of q.
Euler transform of period 12 sequence [ 1, 3, 6, 4, 1, -6, 1, 4, 6, 3, 1, 0, ...].
Convolution of A113973 and A132974. a(n) = A164616(3*n).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015
A128641(n) = (-1)^n*a(n). - Michael Somos, Apr 24 2023

A128640 Expansion of (1/3) * (c(q^2)^2 / c(q)) / (b(q^2)^2 / b(q)) in powers of q where b(), c() are cubic AGM theta functions.

Original entry on oeis.org

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: 1

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. = q - 4*q^2 + 10*q^3 - 20*q^4 + 39*q^5 - 76*q^6 + 140*q^7 - 244*q^8 + ...

Links

Crossrefs

Cf. A128633, A128641, A164617.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^(3/2)] / EllipticTheta[ 2, 0, q^(1/2)])^4, {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( ((eta(x^6 + A) / eta(x^2 + A))^2 * eta(x + A) / eta(x^3 + A))^4, n))};

Formula

Expansion of q * (psi(q^3) / psi(q))^4 in powers of q where psi() is a Ramanujan theta function.
Expansion of ((eta(q^6) / eta(q^2))^2 * (eta(q) / eta(q^3)))^4 in powers of q.
Euler transform of period 6 sequence [ -4, 4, 0, 4, -4, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v * (1-u) * (1-9*u) - (u-v)^2.
G.f.: x * (Product_{k>0} (1 - x^k + x^(2*k))^2 * (1 + x^k + x^(2*k)) )^4.
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (81*u^2*v^2 + 9*u*v - 12*u + 30*u^2 - 108*u^2*v + 1) * v - u^3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (1/9) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128637.
a(n) = - A128641(n) unless n = 0. Convolution inverse of A128633.
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>=1} a(n)/exp(2*Pi*n) = 1/3 + (2/9)*sqrt(3) - (2/9)*sqrt(6)*3^(1/4). - Simon Plouffe, Mar 02 2021

A128636 Expansion of 3 * (b(q^2)^2 / b(q)) / (c(q)^2 / c(q^2)) in powers of q where b(), c() are cubic AGM theta functions.

Original entry on oeis.org

1, 1, -3, 3, 5, -18, 15, 24, -75, 57, 86, -252, 183, 262, -744, 522, 725, -1998, 1365, 1852, -4986, 3336, 4436, -11736, 7719, 10103, -26322, 17067, 22040, -56682, 36306, 46336, -117867, 74700, 94378, -237744, 149277, 186926, -466836, 290706, 361126, -895014
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 - 3*q^2 + 3*q^3 + 5*q^4 - 18*q^5 + 15*q^6 + 24*q^7 - 75*q^8 + ...

Links

Crossrefs

Cf. A123633, A128641.

Programs

  • Mathematica
    eta[x_] := x^(1/24)*QPochhammer[x]; A128636[n_] := SeriesCoefficient[(eta[q^6]/eta[q])*(eta[q^2]/eta[q^3])^5, {q, 0, n}]; Table[A128636[n], {n, 0, 50}] (* G. C. Greubel, Aug 21 2017 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x^3 + A))^5 * eta(x^6 + A) / eta(x + A), n))};

Formula

Expansion of (psi(q)^3 / psi(q^3)) / (phi(-q^3)^3 / phi(-q)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (eta(q^6) / eta(q)) * (eta(q^2) / eta(q^3))^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) = v* (1-v)* (9-8*u) + (u-v)^2.
G.f.: Product_{k>0} (1 - x^(6*k)) / (1 - x^k) * ((1 - x^(2*k)) / (1 - x^(3*k)))^5.
A123633(n) = a(n) unless n = 0. Convolution inverse of A128641.
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = -(3/8)*sqrt(3) + (3/8)*sqrt(9 + 6*sqrt(3)). - Simon Plouffe, Mar 02 2021

A258100 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, May 20 2015

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. A128641, A164616, A182034, A258099.

Programs

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

Formula

Expansion of (psi(q) * f(-q^9)^3) / (chi(-q^3)^2 * psi(q^3)^4) in powers of q where psi(), chi(), f() are Ramanujan theta functions.
Expansion of eta(q^2)^2 * eta(q^3)^2 * eta(q^9)^3 / (eta(q) * eta(q^6)^6) 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 * A164616(n). a(3*n) = A128641(n). a(3*n + 1) = A258099(n). a(3*n + 2) = 0.
Convolution invserse is A182034.

A145977 Expansion of c(q^3) / (c(q^3) + c(q^6)) where c() is a cubic AGM function.

Original entry on oeis.org

1, -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, -20007
Offset: 0

Views

Author

Michael Somos, Oct 26 2008

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^3 + 2*q^4 - 3*q^5 + 4*q^6 - 5*q^7 + 7*q^8 - 10*q^9 + ...

Links

Crossrefs

Cf. A124243, A128129, A128641, A132302, A139032.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 1 - EllipticTheta[ 2, 0, x^(9/2)] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, n}]; (* Michael Somos, Aug 26 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A) * eta(x^9 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^18 + A)), n))};

Formula

Expansion of 1 - q * psi(q^9) / psi(q) = phi(-q^9) / (psi(q) * chi(-q^3)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^6) * eta(q^9)^2 / (eta(q^2)^2 * eta(q^3) * eta(q^18)), in powers of q.
Euler transform of period 18 sequence [ -1, 1, 0, 1, -1, 1, -1, 1, -2, 1, -1, 1, -1, 1, 0, 1, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (2/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A139032.
G.f.: Product_{k>0} (P(3, x^k) * P(9, x^k)) / (P(4, x^k)^2 * P(18, x^k)) where P(n, x) is the n-th cyclotomic polynomial.
Convolution inverse of A139032.
a(n) = - A124243(n) unless n=0. a(2*n) = A128129(n) = a(2*n) unless n=0.
a(2*n + 1) = - A132302(n). a(3*n) = A128641(n).
Showing 1-5 of 5 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.