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.

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A260942 Expansion of x * phi(-x) * psi(x^12) / chi(-x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

0, 1, -2, 0, 1, 0, 0, 1, 0, 0, 0, -2, 0, 1, -2, 0, 2, 0, 0, 1, 0, 0, 1, -2, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0, 1, 0, 0, 2, -2, 0, 1, -2, 0, 1, 0, 0, 1, -2, 0, 1, 0, 0, 1, 0, 0, 0, -2, 0, 2, 0, 0, 0, 0, 0, 0, -2, 0, 1, -2, 0, 0, 0, 0, 2, -2, 0, 1, 0, 0, 3, 0, 0, 1
Offset: 0

Views

Author

Michael Somos, Aug 04 2015

Keywords

Comments

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

Examples

			G.f. = x - 2*x^2 + x^4 + x^7 - 2*x^11 + x^13 - 2*x^14 + 2*x^16 + x^19 + ...
G.f. = q^13 - 2*q^21 + q^37 + q^61 - 2*q^93 + q^109 - 2*q^117 + 2*q^133 + ...

Links

Crossrefs

Cf. A112609, A131963.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^6] QPochhammer[ -x^3, x^3] / (2 x^(1/2)), {x, 0, n}];
eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[ q^(-5/8)* eta[q]^2*eta[q^6]*eta[q^24]^2/(eta[q^2]*eta[q^3]*eta[q^12]), {q, 0, 50}], q] (* G. C. Greubel, Aug 01 2018 *)
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A) * eta(x^24 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))};

Formula

Expansion of q^(-5/8) * eta(q)^2 * eta(q^6) * eta(q^24)^2 / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 24 sequence [ -2, -1, -1, -1, -2, -1, -2, -1, -1, -1, -2, 0, -2, -1, -1, -1, -2, -1, -2, -1, -1, -1, -2, -2, ...].
a(3*n) = 0. a(3*n + 1) = A131963(n). a(3*n + 2) = -2 * A112609(n).

A262726 Expansion of phi(-x) * psi(x^6) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 0, 0, 2, 0, 1, -2, 0, -2, 2, 0, 0, 0, 0, -2, 2, 0, 1, -2, 0, 0, 4, 0, 0, -2, 0, -2, 0, 0, 0, -2, 0, 0, 2, 0, 3, -2, 0, 0, 2, 0, 2, -2, 0, -2, 0, 0, 0, -2, 0, 0, 2, 0, 2, -2, 0, 0, 0, 0, 1, -4, 0, 0, 4, 0, 0, -2, 0, -2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Sep 28 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 + 2*x^4 + x^6 - 2*x^7 - 2*x^9 + 2*x^10 - 2*x^15 + 2*x^16 + ...
G.f. = q^3 - 2*q^7 + 2*q^19 + q^27 - 2*q^31 - 2*q^39 + 2*q^43 - 2*q^63 + ...

Links

Crossrefs

Cf. A112605, A112608, A112609, A138270, A262780.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 4 n + 3, KroneckerSymbol[ -3, #] &]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^3] / (2 x^(3/4)), {x, 0, n}];
a[ n_] := If[ n < 0, 0, Times @@ (Which[ # < 5, Mod[#, 2], Mod[#, 6] == 5, 1 - Mod[#2, 2], True, (#2  + 1) KroneckerSymbol[ 6, #]^#2] & @@@ FactorInteger @ (4 n + 3))]; (* Michael Somos, Oct 01 2015 *)
  • PARI
    {a(n) = if( n<0, 0, (-1)^n * sumdiv(4*n + 3, d, kronecker(-3, d)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^12 + A)^2 / (eta(x^2 + A) * eta(x^6 + A)), n))};
  • PARI
    {a(n) = my(A, p, e); if( n<0, 0, A = factor(4*n + 3); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, p%2, p%6 == 1, (e+1) * if( p%24 == 1 || p%24 == 19, 1, (-1)^e), 1-e%2 )))}; /* Michael Somos, Oct 01 2015 */

Formula

Expansion of q^(-3/4) * eta(q)^2 * eta(q^12)^2 / (eta(q^2) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [-2, -1, -2, -1, -2, 0, -2, -1, -2, -1, -2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = 192^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(n) = (-1)^n * A112605(n). -2 * a(n) = A138270(2*n + 1).
a(2*n) = A112608(n). a(2*n + 1) = -2 * A112609(n). a(3*n + 2) = 0.
a(n) = A262780(2*n + 1). - Michael Somos, Oct 01 2015

A262774 Expansion of psi(x^2) * phi(-x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 0, 1, -2, 0, -2, 1, 0, 0, -2, 0, 0, 3, 0, 2, -2, 0, 0, 2, 0, 1, 0, 0, -2, 2, 0, 0, -2, 0, -2, 1, 0, 2, -4, 0, 0, 0, 0, 0, -2, 0, 0, 3, 0, 0, -2, 0, -2, 2, 0, 2, 0, 0, 0, 4, 0, 1, -2, 0, -2, 2, 0, 0, 0, 0, 0, 0, 0, 4, -2, 0, 0, 1, 0, 0, -4, 0, -2, 2, 0, 0
Offset: 0

Views

Author

Michael Somos, Oct 01 2015

Keywords

Comments

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

Examples

			G.f. = 1 + x^2 - 2*x^3 - 2*x^5 + x^6 - 2*x^9 + 3*x^12 + 2*x^14 - 2*x^15 + ...
G.f. = q + q^9 - 2*q^13 - 2*q^21 + q^25 - 2*q^37 + 3*q^49 + 2*q^57 + ...

Links

Crossrefs

Cf. A112604, A112604, A112606, A112607, A112608, A112609, A131961, A131963.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 4 n + 1, KroneckerSymbol[ -3, #]&]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 4, 0, x^3] / (2 x^(1/4)), {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^2 QPochhammer[ x^4]^2 / ( QPochhammer[ x^2]  QPochhammer[ x^6]), {x, 0, n}];
a[ n_] := If[ n < 0, 0, Times @@ (Which[ # < 5, Mod[#, 2], Mod[#, 6] == 5, 1 - Mod[#2, 2], True, (#2  + 1) KroneckerSymbol[ 6, #]^#2] & @@@ FactorInteger @ (4 n + 1))];
  • PARI
    {a(n) = if( n<0, 0, (-1)^n * sumdiv(4*n + 1, d, kronecker( -3, d)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A)^2 / (eta(x^2 + A) * eta(x^6 + A)), n))};
  • PARI
    {a(n) = my(A, p, e); if( n<0, 0, A = factor(4*n + 1); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, p%2, p%6 == 1, (e+1) * if( p%24 == 1 || p%24 == 19, 1, (-1)^e), 1-e%2 )))};

Formula

Expansion of q^(-1/4) * eta(q^3)^2 * eta(q^4)^2 / (eta(q^2) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 0, 1, -2, -1, 0, 0, 0, -1, -2, 1, 0, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = 192^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(n) = (-1)^n * A112604(n). a(2*n) = A112606(n). a(2*n + 1) = -2 * A112607(n-1). a(3*n + 1) = 0.
a(6*n) = A131961(n). a(6*n + 2) = A112608(n). a(6*n + 3) = -2 * A131963(n). a(6*n + 5) = -2 * A112609(n).
Previous Showing 11-13 of 13 results.

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