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 1-4 of 4 results.

A261445 Expansion of f(x, x^3) * f(x, x^2)^3 in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 4, 9, 14, 16, 18, 21, 28, 36, 38, 40, 36, 43, 52, 54, 62, 56, 72, 74, 72, 81, 64, 88, 90, 98, 100, 72, 110, 112, 126, 133, 104, 126, 108, 136, 144, 112, 148, 144, 158, 144, 144, 183, 172, 180, 182, 152, 162, 194, 196, 198, 160, 216, 216, 180, 224, 189, 230
Offset: 0

Views

Author

Michael Somos, Aug 18 2015

Keywords

Comments

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

Examples

			G.f. = 1 + 4*x + 9*x^2 + 14*x^3 + 16*x^4 + 18*x^5 + 21*x^6 + 28*x^7 + ...
G.f. = q + 4*q^5 + 9*q^9 + 14*q^13 + 16*q^17 + 18*q^21 + 21*q^25 + 28*q^29 + ...

Links

Crossrefs

Cf. A124815, A209613, A260109 A260165, A260301, A263021.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3] / (QPochhammer[ x, x^6] QPochhammer[ x^5, x^6]))^3 EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x]^6 EllipticTheta[ 4, 0, x^3]^3 EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Nov 13 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^3 EllipticTheta[ 4, 0, x^3]^3 / EllipticTheta[ 4, 0, x]^2, {x, 0, n}]; (* Michael Somos, Nov 13 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^6 / (eta(x + A)^4 * eta(x^6 + A)^3), n))};

Formula

Expansion of f(-x^2)^3 * phi(-x^3)^3 / phi(-x)^2 in powers of x where phi(), f() are Ramanujan theta functions.
Expansion of q^(-1/4) * eta(q^2)^5 * eta(q^3)^6 / (eta(q)^4 * eta(q^6)^3) in powers of q.
Euler transform of period 6 sequence [4, -1, -2, -1, 4, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 12^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A260301. - Michael Somos, Nov 13 2015
a(n) = A260109(2*n) = A263021(3*n) = A124815(4*n + 1) = A209613(4*n + 1). - Michael Somos, Nov 13 2015
a(3*n + 1) = 4 * A260165(n). a(3*n + 2) = 9 * A263021(n). - Michael Somos, Nov 13 2015

A266684 Expansion of f(-x) * f(-x^2)^4 / psi(x^3) in powers of x where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -1, -5, 3, 7, 4, 3, -18, -17, -1, 20, 36, -9, -14, -18, -12, 31, 16, -5, -54, -28, 6, 36, 72, 15, -21, -70, 3, 54, 28, -12, -90, -65, -12, 80, 72, 7, -38, -54, 42, 68, 40, 30, -126, -108, 4, 72, 144, -33, -43, -105, -48, 98, 52, 3, -144, -90, 18, 140, 180
Offset: 0

Views

Author

Michael Somos, Jan 02 2016

Keywords

Comments

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

Examples

			G.f. = 1 - x - 5*x^2 + 3*x^3 + 7*x^4 + 4*x^5 + 3*x^6 - 18*x^7 - 17*x^8 + ...

Links

Crossrefs

Cf. A263021, A260301.

Programs

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

Formula

Expansion of eta(q) * eta(q^2)^4 * eta(q^3) / eta(q^6)^2 in powers of q.
Euler transform of period 6 sequence [ -1, -5, -2, -5, -1, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 248832^(1/2) (t/I)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263021.
a(n) = A260301(3*n). a(3*n) = A260301(n).

A279261 Expansion of q^(-1/3) * eta(q)^3 * eta(q^3)^3 / eta(q^2)^2 in powers of q.

Original entry on oeis.org

1, -3, 2, -4, 14, -11, 6, -20, 21, -14, 10, -16, 38, -20, 14, -40, 43, -42, 16, -28, 62, -43, 22, -40, 74, -42, 26, -40, 64, -68, 28, -80, 98, -63, 34, -52, 110, -62, 32, -100, 133, -70, 42, -56, 108, -80, 46, -120, 112, -114, 50, -72, 158, -84, 54, -140, 183
Offset: 0

Views

Author

Michael Somos, Dec 08 2016

Keywords

Examples

			G.f. = 1 - 3*x + 2*x^2 - 4*x^3 + 14*x^4 - 11*x^5 + 6*x^6 - 20*x^7 + ...
G.f. = q - 3*q^4 + 2*q^7 - 4*q^10 + 14*q^13 - 11*q^16 + 6*q^19 + ...

Links

Crossrefs

Cf. A260301.

Programs

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

Formula

Euler transform of period 6 sequence [ -3, -1, -6, -1, -3, -4, ...].
3 * a(n) = A260301(3*n + 1).

A279270 Expansion of phi(-x) * chi(-x)^2 * f(-x^6)^3 in powers of x where phi(), chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -4, 5, -4, 10, -16, 12, -8, 14, -28, 21, -8, 30, -40, 28, -16, 21, -52, 34, -20, 50, -56, 48, -24, 38, -72, 44, -28, 70, -88, 56, -24, 43, -100, 70, -36, 80, -112, 84, -32, 62, -104, 85, -44, 110, -136, 56, -56, 74, -148, 102, -40, 130, -144, 120, -56, 64
Offset: 0

Views

Author

Michael Somos, Dec 09 2016

Keywords

Comments

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

Examples

			G.f. = 1 - 4*x + 5*x^2 - 4*x^3 + 10*x^4 - 16*x^5 + 12*x^6 - 8*x^7 + ...
G.f. = q^2 - 4*q^5 + 5*q^8 - 4*q^11 + 10*q^14 - 16*q^17 + 12*q^20 + ...

Links

Crossrefs

Cf. A260301.

Programs

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

Formula

Expansion of q^(-2/3) * eta(q)^4 * eta(q^6)^3 / eta(q^2)^3 in powers of q.
Euler transform of period 6 sequence [ -4, -1, -4, -1, -4, -4, ...].
3 * a(n) = A260301(3*n + 2).
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.