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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A260943 Expansion of psi(-x^2) * chi(x^3) * f(x^6) in powers of x where psi(), chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 0, -1, 1, 0, -1, 0, 0, -1, 1, 0, -2, 0, 0, 0, 0, 0, -1, 3, 0, -1, 2, 0, 0, 0, 0, -1, 2, 0, 0, 1, 0, -2, 0, 0, -1, 2, 0, -1, 0, 0, -1, 0, 0, 0, 1, 0, -1, 2, 0, -2, 0, 0, -2, 0, 0, 0, 0, 0, -1, 0, 0, -1, 3, 0, -1, 0, 0, -1, 0, 0, -1, 2, 0, 0, 2, 0, -1, 0, 0
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. = 1 - x^2 + x^3 - x^5 - x^8 + x^9 - 2*x^11 - x^17 + 3*x^18 - x^20 + ...
G.f. = q^3 - q^19 + q^27 - q^43 - q^67 + q^75 - 2*q^91 - q^139 + 3*q^147 + ...

Links

Crossrefs

Cf. A112606, A131964.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x] QPochhammer[ -x^6] QPochhammer[ -x^3, x^6] / (2^(1/2) x^(1/4)), {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A) * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x^3 + A) * eta(x^4 + A) * eta(x^24 + A)), n))};

Formula

Expansion of q^(-3/8) * eta(q^2) * eta(q^6) * eta(q^8) * eta(q^12)^2 / (eta(q^3) * eta(q^4) * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [ 0, -1, 1, 0, 0, -1, 0, -1, 1, -1, 0, -2, 0, -1, 1, -1, 0, -1, 0, 0, 1, -1, 0, -2, ...].
a(3*n) = A112606(n). a(3*n + 1) = 0. a(3*n + 2) = - A131964(n).

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 11-12 of 12 results.

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