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.

A201078 Twice A137829.

Original entry on oeis.org

2, 4, 12, 24, 50, 92, 172, 296, 510, 840, 1372, 2176, 3424, 5268, 8040, 12072, 17976, 26428, 38564, 55680, 79846, 113500, 160320, 224768, 313320, 433916, 597788, 818840, 1116238, 1513900, 2044180, 2747520, 3677864, 4902732, 6510960, 8613840, 11356208
Offset: 0

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Author

N. J. A. Sloane, Nov 26 2011

Keywords

Links

A137828 Expansion of phi(x) / f(-x^4)^2 in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 0, 0, 4, 4, 0, 0, 9, 12, 0, 0, 20, 24, 0, 0, 42, 50, 0, 0, 80, 92, 0, 0, 147, 172, 0, 0, 260, 296, 0, 0, 445, 510, 0, 0, 744, 840, 0, 0, 1215, 1372, 0, 0, 1944, 2176, 0, 0, 3059, 3424, 0, 0, 4740, 5268, 0, 0, 7239, 8040, 0, 0, 10920, 12072, 0, 0, 16286
Offset: 0

Views

Author

Michael Somos, Feb 12 2008

Keywords

Comments

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

Examples

			G.f. = 1 + 2*x + 4*x^4 + 4*x^5 + 9*x^8 + 12*x^9 + 20*x^12 + 24*x^13 + 42*x^16 + ...
G.f. = 1/q + 2*q^2 + 4*q^11 + 4*q^14 + 9*q^23 + 12*q^26 + 20*q^35 + 24*q^38 + ...

Links

Crossrefs

Cf. A051136, A137829.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] / QPochhammer[ x^4]^2, {x, 0, n}]; (* Michael Somos, Oct 04 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / eta(x^4 + A)^4 / eta(x + A)^2, n))};

Formula

Expansion of q^(1/3) * eta(q^2)^5 / (eta(q)^2 * eta(q^4)^4) in powers of q.
Euler transform of period 4 sequence [ 2, -3, 2, 1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 6^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A051136.
G.f.: Product_{k>0} (1 + x^k) / ( (1 - x^k) * (1 + x^(2*k))^4 ).
a(4*n + 2) = a(4*n + 3) = 0.
a(4*n) = A051136(n). a(4*n + 1) = 2 * A137829(n).

A137830 Expansion of phi(-x) / f(-x^4)^2 in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 0, 0, 4, -4, 0, 0, 9, -12, 0, 0, 20, -24, 0, 0, 42, -50, 0, 0, 80, -92, 0, 0, 147, -172, 0, 0, 260, -296, 0, 0, 445, -510, 0, 0, 744, -840, 0, 0, 1215, -1372, 0, 0, 1944, -2176, 0, 0, 3059, -3424, 0, 0, 4740, -5268, 0, 0, 7239, -8040, 0, 0, 10920
Offset: 0

Views

Author

Michael Somos, Feb 12 2008

Keywords

Comments

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

Examples

			G.f. = 1 - 2*x + 4*x^4 - 4*x^5 + 9*x^8 - 12*x^9 + 20*x^12 - 24*x^13 + 42*x^16 + ...
G.f. = 1/q - 2*q^2 + 4*q^11 - 4*q^14 + 9*q^23 - 12*q^26 + 20*q^35 - 24*q^38 + ...

Links

Crossrefs

Cf. A051136, A137828, A137829.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] / QPochhammer[ x^4]^2, {x, 0, n}]; (* Michael Somos, Oct 04 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 / eta(x^2 + A) / eta(x^4 + A)^2, n))};

Formula

Expansion of q^(1/3) * eta(q)^2 / (eta(q^2) * eta(q^4)^2) in powers of q.
Euler transform of period 4 sequence [ -2, -1, -2, 1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = (4/3)^(1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A137829.
G.f.: ( Product_{k>0} (1 - x^(2*k)) * (1 + x^k)^2 * (1 + x^(2*k))^2 )^(-1).
a(4*n + 2) = a(4*n + 3) = 0.
a(n) = (-1)^n * A137828(n). a(4*n) = A051136(n). a(4*n + 1) = -2 * A137829(n).
Showing 1-3 of 3 results.

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