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.

A330311 Where n appears in A329544, or -1 if n never appears.

Original entry on oeis.org

1, 3, 2, 5, 4, 9, 13, 12, 45, 40, 7, 56, 18, 11, 14, 15, 10, 19, 6, 44, 43, 8, 55, 17, 28, 21, 16, 25, 20, 29, 63, 42, 23, 175, 34, 27, 22, 31, 26, 35, 30, 80, 75, 24, 39, 33, 48, 37, 32, 49, 36, 85, 79, 76, 46, 187, 130, 47, 38, 51, 50, 89, 84, 110, 109
Offset: 1

Views

Author

N. J. A. Sloane, Dec 10 2019

Keywords

Comments

a(919) is unknown. If it is not -1, it is greater than 10^6 (see A329544).

Links

Crossrefs

Cf. A104444, A329544, A329797, A328798.

Formula

a(A104444(n)) = -1. - Rémy Sigrist, Dec 11 2019

A328795 Expansion of (chi(x) * chi(-x^3))^2 in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 1, 0, 0, 2, 2, 0, 2, 2, 1, 0, 2, 6, 2, 0, 3, 6, 4, 0, 4, 8, 4, 0, 7, 14, 7, 0, 6, 16, 10, 0, 11, 20, 11, 0, 14, 32, 16, 0, 17, 38, 21, 0, 22, 46, 24, 0, 32, 66, 34, 0, 34, 78, 44, 0, 49, 96, 50, 0, 60, 130, 66, 0, 72, 154, 84, 0, 90, 186, 98, 0, 117, 244
Offset: 0

Views

Author

Michael Somos, Oct 28 2019

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution square of A328802.
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A328789.

Examples

			G.f. = 1 + 2*x + x^2 + 2*x^5 + 2*x^6 + 2*x^8 + 2*x^9 + x^10 + ...
G.f. = q^-1 + 2*q^2 + q^5 + 2*q^14 + 2*q^17 + 2*q^23 + 2*q^26 + ...

Links

Crossrefs

Cf. A112206, A328789, A328790, A328797, A328798, A328802.

Programs

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

Formula

Euler transform of period 12 sequence [2, -2, 0, 0, 2, -2, 2, 0, 0, -2, 2, 0, ...].
G.f.: Product_{k>=1} (1 + x^(2*k-1))^2 * (1 - x^(6*k-3))^2.
a(n) = (-1)^n * A328797(n). a(2*n) = A112206(n).
a(4*n) = A328789(n). a(4*n + 1) = 2 * A328798(n). a(4*n + 2) = A328790(n). a(4*n + 3) = 0.

A328797 Expansion of (chi(-x) * chi(x^3))^2 in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 1, 0, 0, -2, 2, 0, 2, -2, 1, 0, 2, -6, 2, 0, 3, -6, 4, 0, 4, -8, 4, 0, 7, -14, 7, 0, 6, -16, 10, 0, 11, -20, 11, 0, 14, -32, 16, 0, 17, -38, 21, 0, 22, -46, 24, 0, 32, -66, 34, 0, 34, -78, 44, 0, 49, -96, 50, 0, 60, -130, 66, 0, 72, -154, 84, 0, 90, -186
Offset: 0

Views

Author

Michael Somos, Oct 27 2019

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution square of A328800.
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A328790.

Examples

			G.f. = 1 - 2*x + x^2 - 2*x^5 + 2*x^6 + 2*x^8 - 2*x^9 + x^10 + ...
G.f. = q^-1 - 2*q^2 + q^5 - 2*q^14 + 2*q^17 + 2*q^23 - 2*q^26 + ..

Links

Crossrefs

Cf. A112206, A328789, A328790, A328795, A328798, A328800.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ x, x^2] QPochhammer[ -x^3, x^6])^2, {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)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A))^2, n))};

Formula

Expansion of q^(1/3) * (eta(q) * eta(q^6)^2)^2 / (eta(q^2) * eta(q^3) * eta(q^12))^2 in powers of q.
Euler transform of period 12 sequence [-2, 0, 0, 0, -2, -2, -2, 0, 0, 0, -2, 0, ...].
G.f.: Product_{k>=1} (1 - x^(2*k-1))^2 * (1 + x^(6*k-3))^2.
a(n) = (-1)^n * A328795(n). a(2*n) = A112206(n).
a(4*n) = A328789(n). a(4*n + 1) = -2 * A328798(n). a(4*n + 2) = A328790(n). a(4*n + 3) = 0.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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