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.

A213627 Expansion of psi(x)^4 / psi(x^3) in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 4, 6, 7, 9, 6, 7, 15, 12, 12, 13, 6, 12, 18, 18, 13, 15, 18, 12, 24, 12, 13, 27, 12, 24, 15, 12, 24, 28, 30, 12, 27, 18, 12, 30, 18, 19, 27, 24, 24, 27, 24, 36, 30, 18, 19, 24, 24, 24, 45, 18, 12, 45, 30, 24, 28, 18, 36, 36, 36, 24, 15, 36, 36, 51, 18, 25
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).

Examples

			G.f. = 1 + 4*x + 6*x^2 + 7*x^3 + 9*x^4 + 6*x^5 + 7*x^6 + 15*x^7 + 12*x^8 + ...
G.f. = q + 4*q^9 + 6*q^17 + 7*q^25 + 9*q^33 + 6*q^41 + 7*q^49 + 15*q^57 + 12*q^65 + ...

Links

Crossrefs

Cf. A212907.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      add([-3, 4, -4, 3, -4, 4][1+irem(d, 6)]*d,
        d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 18 2020
  • Mathematica
    a[ n_] := SeriesCoefficient[ 1/8 EllipticTheta[ 2, 0, q]^4 / EllipticTheta[ 2, 0, q^3], {q, 0, 2 n + 1/4}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 * eta(x^3 + A) / (eta(x + A)^4 * eta(x^6 + A)^2), n))};

Formula

Expansion of q^(-1/8) * eta(q^2)^8 * eta(q^3) / (eta(q)^4 * eta(q^6)^2) in powers of q.
a(3*n + 2) = 6 * A212907(n).
Euler transform of period 6 sequence [4, -4, 3, -4, 4, -3, ...]. - Georg Fischer, Aug 18 2020

A263452 Expansion of f(-q^3)^3 * psi(q^12) / f(-q) in powers of q where ps(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 2, 0, 2, 1, 2, 0, 1, 2, 2, 0, 3, 1, 4, 0, 5, 3, 2, 0, 3, 3, 4, 0, 4, 2, 4, 0, 3, 2, 4, 0, 4, 2, 4, 0, 5, 5, 4, 0, 3, 3, 8, 0, 7, 3, 6, 0, 4, 4, 4, 0, 6, 4, 4, 0, 9, 3, 6, 0, 4, 4, 4, 0, 4, 3, 8, 0, 5, 5, 6, 0, 9, 3, 4, 0, 7, 6, 6, 0, 7, 6, 10, 0, 6, 3
Offset: 0

Views

Author

Michael Somos, Oct 18 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^2 + 2*x^4 + x^5 + 2*x^6 + x^8 + 2*x^9 + 2*x^10 + ...
G.f. = q^11 + q^17 + 2*q^23 + 2*q^35 + q^41 + 2*q^47 + q^59 + 2*q^65 + ...

Links

Crossrefs

Cf. A212907, A261444, A263527.

Programs

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

Formula

Expansion of q^(-11/6) * eta(q^3)^3 * eta(q^24)^2 / (eta(q) * eta(q^12)) in powers of q.
Euler transform of period 24 sequence [ 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -1, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -3, ...].
2 * a(n) = A261444(2*n + 1). a(4*n + 1) = A212907(n). a(4*n + 3) = 0.
-2 * a(n) = A263527(2*n + 3). - Michael Somos, Nov 05 2015

A263501 Expansion of phi(-x) * f(-x^2)^3 / f(-x^3) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, -3, 7, 0, -3, 7, -12, -6, 12, -2, -3, 12, 0, -9, 13, -12, -9, 12, -12, -6, 13, 0, -6, 24, -12, -6, 24, -14, -15, 12, 0, -9, 12, -24, -9, 19, 0, -12, 24, 0, -12, 36, -24, -9, 19, -12, -12, 24, 0, -9, 12, -36, -15, 24, -14, -9, 36, 0, -18, 24, -12, -18
Offset: 0

Views

Author

Michael Somos, Oct 19 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*x^2 + 7*x^3 - 3*x^5 + 7*x^6 - 12*x^7 - 6*x^8 + ...
G.f. = q - 2*q^9 - 3*q^17 + 7*q^25 - 3*q^41 + 7*q^49 - 12*q^57 + ...

Links

Crossrefs

Cf. A212907, A263456, A263527.

Programs

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

Formula

Expansion of q^(-1/8) * eta(q)^2 * eta(q^2)^2 / eta(q^3) in powers of q.
Euler transform of period 6 sequence [ -2, -4, -1, -4, -2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 4374^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263527.
-2 * a(n) = A263456(8*n + 1). a(3*n + 2) = -3 * A212907(n). a(9*n + 4) = 0.
Showing 1-3 of 3 results.

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