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.

A213618 Expansion of phi(-q^3) * b(q^8) in powers of q where phi() is a Ramanujan theta function and b() is a cubic AGM theta function.

Original entry on oeis.org

1, 0, 0, -2, 0, 0, 0, 0, -3, 0, 0, 6, 2, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 6, 0, 0, -14, 0, 0, 0, 0, -3, 0, 0, 12, 12, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 2, 0, 0, -12, 0, 0, 0, 0, -12, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, -12, 0, 0, 0, 18, 0, 0, -14, 0, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Jun 16 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 - 2*q^3 - 3*q^8 + 6*q^11 + 2*q^12 - 6*q^20 + 6*q^24 - 14*q^27 + ...

Links

Crossrefs

Cf. A014452, A213607, A213617.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3] QPochhammer[ q^8]^3 / QPochhammer[ q^24], {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^8 + A)^3 / (eta(x^6 + A) * eta(x^24 + A)), n))};

Formula

Expansion of eta(q^3)^2 * eta(q^8)^3 / (eta(q^6) * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [ 0, 0, -2, 0, 0, -1, 0, -3, -2, 0, 0, -1, 0, 0, -2, -3, 0, -1, 0, 0, -2, 0, 0, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 93312^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A213607.
a(3*n + 1) = a(4*n + 1) = a(4*n + 2) = a(24*n + 15) = a(24*n + 23) = 0.
a(12*n) = A014452(n). a(24*n + 8) = -3 * A213592(n). a(24*n + 11) = 6 * A213617(n). a(24*n + 20) = -6 * A213607(n).

A298932 Expansion of f(-x^3)^3 * phi(-x^12) / (f(-x) * chi(-x^4)) in powers of x where phi(), chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 2, 0, 3, 2, 4, 0, 4, 4, 6, 0, 5, 3, 6, 0, 6, 4, 4, 0, 8, 4, 6, 0, 9, 6, 6, 0, 6, 6, 12, 0, 8, 4, 12, 0, 8, 7, 8, 0, 9, 6, 8, 0, 12, 8, 6, 0, 8, 6, 14, 0, 12, 6, 12, 0, 8, 8, 12, 0, 13, 6, 12, 0, 18, 10, 8, 0, 8, 12, 12, 0, 16, 7, 14, 0, 12, 8, 12, 0, 16
Offset: 0

Views

Author

Michael Somos, Jan 29 2018

Keywords

Comments

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

Examples

			G.f. = 1 + x + 2*x^2 + 3*x^4 + 2*x^5 + 4*x^6 + 4*x^8 + 4*x^9 + ...
G.f. = q + q^3 + 2*q^5 + 3*q^9 + 2*q^11 + 4*q^13 + 4*q^17 + 4*q^19 + ...

Links

Crossrefs

Cf. A213607, A298931, A298933.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^3 QPochhammer[ -x^4, x^4] EllipticTheta[ 4, 0, x^12] / QPochhammer[ x], {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^24 + A)), n))};

Formula

Expansion of q^(-1/2) * eta(q^3)^3 * eta(q^8) * eta(q^12)^2 / (eta(q) * eta(q^4) * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [1, 1, -2, 2, 1, -2, 1, 1, -2, 1, 1, -3, 1, 1, -2, 1, 1, -2, 1, 2, -2, 1, 1, -3, ...].
a(4*n + 3) = 0. a(3*n + 2) = 2 * A213607(n). a(n) = A298931(3*n). a(2*n) = A298933(n).

A245668 Expansion of (chi(q^3) * psi(-q))^3 in powers of q where chi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -3, 3, -1, -3, 6, -3, 0, 3, 3, -12, 6, -1, -12, 12, 0, -3, 12, 9, -12, 6, -6, -12, 0, -3, -15, 18, 5, 0, 18, -6, 0, 3, -6, -24, 12, 3, -12, 18, 0, -12, 24, -6, -12, 6, 18, -24, 0, -1, -27, 21, -6, -12, 18, 15, 0, 12, -6, -12, 18, 0, -36, 24, 0, -3, 24, -12
Offset: 0

Views

Author

Michael Somos, Jul 28 2014

Keywords

Comments

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

Examples

			G.f. = 1 - 3*q + 3*q^2 - q^3 - 3*q^4 + 6*q^5 - 3*q^6 + 3*q^8 + 3*q^9 + ...

Links

Crossrefs

Cf. A089807, A213592, A213607, A245669.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(12), 3/2), 67);  A[1] - 3*A[2] + 3*A[3];
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[3, Pi/3, q]^3, {q,0,n}];
a[ n_] := SeriesCoefficient[ ((3 EllipticTheta[3, 0, q^9] - EllipticTheta[3, 0, q]) / 2)^3, {q,0,n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[-q^3, q^6] EllipticTheta[2, 0, Sqrt[-q]] / (2 (-q)^(1/8)))^3, {q,0,n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)))^3, n))};

Formula

Expansion of phi(q^3) * psi(-q)^3 / psi(-q^3) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2) * eta(q^3) * eta(q^12)))^3 in powers of q.
Euler transform of period 12 sequence [-3, 0, 0, -3, -3, -3, -3, -3, 0, 0, -3, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g(t) is g.f. for A245669.
a(3*n + 1) = -3 * A213056(n). a(6*n + 2) = 3 * A213592(n). a(6*n + 5) = 6 * A213607(n). a(8*n + 7) = 0.
Convolution cube of A089807.

A257651 Expansion of chi(x)^2 * f(-x^6)^3 in powers of x where chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 1, 2, 4, 4, 2, 0, 6, 6, 1, 4, 6, 8, 2, 0, 7, 6, 4, 6, 8, 8, 4, 0, 10, 8, 2, 6, 10, 12, 0, 0, 9, 14, 6, 6, 12, 8, 6, 0, 10, 12, 1, 10, 14, 8, 4, 0, 16, 14, 6, 8, 8, 16, 8, 0, 12, 14, 2, 10, 12, 16, 0, 0, 20, 10, 7, 8, 20, 20, 6, 0, 10, 16, 4, 10, 20, 12
Offset: 0

Views

Author

Michael Somos, Jul 25 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 + 2*x + x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 2*x^6 + 6*x^8 + 6*x^9 + ...
G.f. = q^2 + 2*q^5 + q^8 + 2*q^11 + 4*q^14 + 4*q^17 + 2*q^20 + 6*q^26 + ...

Links

Crossrefs

Cf. A213023, A213592, A213607, A213617.

Programs

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

Formula

Expansion of phi(x^3) * f(x, x^5)^2 in powers of x where phi(), f() are Ramanujan theta functions.
Expansion of phi(x) * c(x^2) / 3 in powers of x where phi() is a Ramanujan theta function and c() is a cubic AGM theta function.
Expansion of q^(-2/3) * eta(q^2)^4 * eta(q^6)^3 / (eta(q)^2 * eta(q^4)^2) in powers of q.
Euler transform of period 12 sequence [ 2, -2, 2, 0, 2, -5, 2, 0, 2, -2, 2, -3, ...].
a(n) = a(4*n + 2) = A213592(2*n + 1). a(2*n) = A213592(n). a(2*n + 1) = 2 * A213607(n).
a(8*n + 2) = A213592(n). a(8*n + 3) = 2 * A213617(n). a(8*n + 5) = 4 * A213023(n). a(8*n + 6) = 2 * A213607(n). a(8*n + 7) = 0.
Showing 1-4 of 4 results.

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