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 1-3 of 3 results.

A255258 Expansion of q^2 * phi(q) * psi(q^16) in powers of q where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Views

Author

Michael Somos, Feb 19 2015

Keywords

Comments

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

Examples

			G.f. = q^2 + 2*q^3 + 2*q^6 + 2*q^11 + 3*q^18 + 2*q^19 + 2*q^22 + 4*q^27 + ...

Links

Crossrefs

Cf. A033761, A113411, A224609, A227395.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(32), 1), 89); A[3] + 2*A[4] + 2*A[7] + 2*A[12];
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q^8] / 2, {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^32 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^16 + A)), n))};

Formula

Expansion of eta(q^2)^5 * eta(q^32)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^16)) in powers of q.
Euler transform of period 32 sequence [ 2, -3, 2, -1, 2, -3, 2, -1, 2, -3, 2, -1, 2, -3, 2, 0, 2, -3, 2, -1, 2, -3, 2, -1, 2, -3, 2, -1, 2, -3, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 8^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A224609.
(-1)^n * a(n) = A227395(n).
a(4*n) = a(4*n + 1) = a(8*n + 7) = 0. a(4*n + 2) = A113411(n). a(8*n + 3) = 2 * A033761(n).

A255320 Expansion of chi(-x) * psi(x^3) * psi(x^48) in powers of x where chi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Views

Author

Michael Somos, Feb 21 2015

Keywords

Comments

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

Examples

			G.f. = 1 - x - x^5 + x^8 + x^16 - x^21 - x^33 + x^40 + x^48 - x^49 + ...
G.f. = q^19 - q^22 - q^34 + q^43 + q^67 - q^82 - q^118 + q^139 + q^163 + ...

Links

Crossrefs

Cf. A227395, A255317, A255318, A255319, A256574.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] EllipticTheta[ 2, 0, x^(3/2)] EllipticTheta[2, 0, x^(24)] / (4 x^(51/8)), {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^2 * eta(x^96 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^48 + A)), n))};
  • PARI
    {a(n) = my(A, p, e); if( n<0 || n%8 == 2, 0, A = factor(3*n + 19); 1/2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, -(p+e==3), p%8 > 4, 1-e%2, e+1)))}; /* Michael Somos, Apr 24 2015 */

Formula

Expansion of q^(-19/3) * eta(q) * eta(q^6)^2 * eta(q^96)^2 / (eta(q^2) * eta(q^3) * eta(q^48)) in powers of q.
Euler transform of a period 96 sequence.
a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = a(16*n + 9) = a(16*n + 13) = 0.
-2 * a(n) = A227395(3*n + 19). a(8*n) = A255317(n). a(16*n + 1) = -A255318(n). a(16*n + 5) = -A255319(n).
a(n) = (-1)^n * A256574(n). - Michael Somos, Apr 24 2015

A255365 Expansion of phi(-x^3) * phi(-x^48) / chi(-x^16) in powers of x where phi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Views

Author

Michael Somos, Feb 21 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^3 + 2*x^12 + x^16 - 2*x^19 - 2*x^27 + 2*x^28 + x^32 + ...
G.f. = q^2 - 2*q^11 + 2*q^38 + q^50 - 2*q^59 - 2*q^83 + 2*q^86 + q^98 + ...

Links

Crossrefs

Cf. A002448, A227395.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ -x^16, x^16] EllipticTheta[ 4, 0, x^3] EllipticTheta[ 3, 0, x^48], {x, 0, n}];
a[n_]:=SeriesCoefficient[EllipticTheta[3,0,-x^3]*EllipticTheta[3,0,-x^48 ]/QPochhammer[x^16, x^32], {x, 0, n}]; Table[a[n], {n,0,100}] (* G. C. Greubel, Mar 14 2018 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^32 + A) * eta(x^48 + A)^2 / (eta(x^6 + A) * eta(x^16 + A) * eta(x^96 + A)), n))};
  • PARI
    {a(n) = my(A, p, e); if( n<0 || n%4==1, 0, n = 3*n+2; A = factor(n); - prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, -(e==1), if( p==3, 0, if( p%8 < 4, e+1, 1-e%2))))))};

Formula

Expansion of q^(-2/3) * eta(q^3)^2 * eta(q^32) * eta(q^48)^2 / (eta(q^6) * eta(q^16) * eta(q^96)) in powers of q.
Euler transform of a period 96 sequence.
a(4*n + 1) = a(4*n + 2) = a(8*n + 7) = a(16*n + 4) = a(16*n + 8) = 0.
a(3*n) = A002448(n). a(3*n + 32) = A227395(n).
Showing 1-3 of 3 results.

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