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.

A132179 Expansion of f(-x^2)^2 * f(x, x^2) / f(-x^3)^3 in powers of x where f(,) is a Ramanujan theta function.

Original entry on oeis.org

1, 1, -1, 1, 0, -3, 4, 1, -6, 5, 1, -10, 11, 4, -19, 17, 4, -31, 31, 9, -50, 46, 11, -79, 77, 21, -122, 112, 28, -183, 173, 46, -273, 249, 62, -396, 370, 98, -573, 521, 130, -815, 751, 193, -1149, 1041, 261, -1599, 1461, 373, -2214, 1998, 498, -3031, 2750, 696, -4125, 3708, 923, -5567
Offset: 0

Views

Author

Michael Somos, Aug 12 2007

Keywords

Comments

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

Examples

			G.f. = 1 + x - x^2 + x^3 - 3*x^5 + 4*x^6 + x^7 - 6*x^8 + 5*x^9 + x^10 + ...
G.f. = 1/q + q^5 - q^11 + q^17 - 3*q^29 + 4*q^35 + q^41 - 6*q^47 + 5*q^53 + ...

Links

Crossrefs

Cf. A058487, A062242, A062244, A092848, A132180, A132301, A182036, A230256, A233037, A254525.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^3 / (QPochhammer[ x]  QPochhammer[ x^3] QPochhammer[ x^6]), {x, 0, n}]; (* Michael Somos, Feb 05 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 / (eta(x + A) * eta(x^3 + A) * eta(x^6 + A)), n))};

Formula

Expansion of (chi(-x) / chi(-x^3)^3) * (psi(x) / psi(x^3))^2 in powers of x where chi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 05 2015
Expansion of q^(1/6) * eta(q^2)^3 / ( eta(q) * eta(q^3) * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ 1, -2, 2, -2, 1, 0, ...].
Given g.f. A(x), then B(q) = A(q^6)/q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u^2 - 3*v)^3 - 4*(u^2*v^2 - v^3)*(u^2*v^2 - 2*v^3).
G.f.: Product_{k>0} (1 + x^k)^2 / ( (1 - x^k + x^(2*k)) * (1 + x^k + x^(2*k))^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132180.
Convolution of A092848 and A058487. - Michael Somos, Feb 05 2015
a(n) = (-1)^n * A254525(n) = A062242(2*n) = A062244(2*n) = A132301(2*n) = A182036(3*n). - Michael Somos, Feb 05 2015
a(2*n) = A230256(n). a(2*n + 1) = A233037(n). - Michael Somos, Feb 05 2015

A132302 Expansion of f(-x, -x^5) * f(-x^6) / f(-x)^2 in powers of x where f(, ) and f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 3, 5, 10, 15, 26, 39, 63, 92, 140, 201, 295, 415, 591, 818, 1140, 1554, 2126, 2861, 3855, 5126, 6816, 8970, 11793, 15372, 20007, 25857, 33356, 42771, 54734, 69683, 88530, 111968, 141312, 177642, 222842, 278557, 347484, 432095, 536230, 663549, 819504
Offset: 0

Views

Author

Michael Somos, Aug 17 2007

Keywords

Comments

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

Examples

			G.f. = 1 + x + 3*x^2 + 5*x^3 + 10*x^4 + 15*x^5 + 26*x^6 + 39*x^7 + ...
G.f. = q + q^3 + 3*q^5 + 5*q^7 + 10*q^9 + 15*q^11 + 26*q^13 + 39*q^15 + ...

Links

Crossrefs

Cf. A124243, A132180, A132301, A132975, A213267.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^6] QPochhammer[ x^5, x^6] QPochhammer[ x^6]^2 / QPochhammer[ x]^2, {x, 0, n}]; (* Michael Somos, Nov 01 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^3 / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A)), n))};

Formula

Expansion of q^(-1/2) * eta(q^6)^3 / (eta(q) * eta(q^2) * eta(q^3)) in powers of q.
Euler transform of period 6 sequence [ 1, 2, 2, 2, 1, 0, ...].
Given g.f. A(x), then B(q) = A(q^2) * q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u^2 - v)^3 - 4 * v^4 * (v - 3*u^2) * (2*v - 3*u^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/6) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132301.
a(n) = A124243(2*n + 1) = A132180(2*n + 1) = A132975(2*n + 1) = A213267(2*n + 1). - Michael Somos, Nov 01 2015
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(7/4)*3^(3/2)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A254372 Expansion of phi(q) * phi(-q^3) * f(-q^12) / f(-q^4)^3 in powers of q where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 0, -2, 1, 6, 0, -10, 3, 20, 0, -30, 1, 52, 0, -78, 6, 126, 0, -184, 3, 280, 0, -402, 12, 590, 0, -830, 5, 1182, 0, -1636, 21, 2280, 0, -3108, 10, 4252, 0, -5722, 36, 7710, 0, -10252, 15, 13632, 0, -17940, 60, 23586, 0, -30744, 26, 40014, 0, -51714, 96
Offset: 0

Views

Author

Michael Somos, Jan 29 2015

Keywords

Comments

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

Examples

			G.f. = 1 + 2*q - 2*q^3 + q^4 + 6*q^5 - 10*q^7 + 3*q^8 + 20*q^9 + ...

Links

Crossrefs

Cf. A132180, A254346, A230256.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] * EllipticTheta[ 4, 0, q^3] QPochhammer[ q^12] / QPochhammer[ q^4]^3, {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^2 * eta(x^12 + A) / (eta(x + A)^2 * eta(x^4 + A)^5 * eta(x^6 + A)), n))};

Formula

Expansion of eta(q^2)^5 * eta(q^3)^2 * eta(q^12) / (eta(q)^2 * eta(q^4)^5 * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 2, -3, 0, 2, 2, -4, 2, 2, 0, -3, 2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = (4/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A230256.
a(4*n + 2) = 0. a(2*n + 1) = 2 * A254346(n). a(4*n) = A132180(n).
Showing 1-3 of 3 results.

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