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.

A258831 Expansion of (psi(-x^3) * f(-x, x^2))^2 in powers of x where psi(), f(,) are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 3, -4, 5, -8, 7, -8, 9, -10, 14, -12, 16, -14, 15, -20, 17, -18, 19, -24, 26, -22, 23, -28, 25, -32, 32, -28, 29, -30, 38, -32, 33, -40, 40, -44, 42, -38, 39, -40, 57, -42, 43, -44, 45, -62, 47, -56, 49, -56, 62, -52, 53, -60, 64, -68, 64, -58, 59, -60
Offset: 0

Views

Author

Michael Somos, Jun 11 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 - 4*x^3 + 5*x^4 - 8*x^5 + 7*x^6 - 8*x^7 + 9*x^8 + ...
G.f. = q^5 - 2*q^11 + 3*q^17 - 4*q^23 + 5*q^29 - 8*q^35 + 7*q^41 - 8*q^47 + ...

Links

Crossrefs

Cf. A098098, A121613, A208435, A208457, A258832.

Programs

  • GAP
    List([0..70], n -> (-1)^n*Sigma(6*n+5)/6); # Muniru A Asiru, Jan 30 2018
  • Magma
    [(-1)^n*SumOfDivisors(6*n+5)/6: n in [0..70]]; // Vincenzo Librandi, Jan 30 2018
  • Maple
    with(numtheory):
seq((-1)^(n-1)*sigma(6*n - 1)/6, n=1..10^3); # Muniru A Asiru, Jan 30 2018
  • Mathematica
    a[ n_] := If[ n < 0, 0, (-1)^n DivisorSigma[ 1, 6 n + 5] / 6];
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^6]^2 QPochhammer[ x, -x] / QPochhammer[ x^3, -x^3])^2, {x, 0, n}];
Table[(-1)^n DivisorSigma[1, 6 n + 5] / 6, {n, 0, 60}] (* Vincenzo Librandi, Jan 30 2018 *)
  • PARI
    {a(n) = if(n<0, 0, (-1)^n*sigma(6*n+5)/6)};
  • PARI
    {a(n) = my(A); if(n<0, 0, A = x*O(x^n); polcoeff((eta(x + A)*eta(x^4 + A)*eta(x^6 + A)^4/(eta(x^2+A)^2*eta(x^3+A)*eta(x^12+A)))^2, n))};

Formula

Expansion of (f(-x^6)^2 * chi(x^3) / chi(x))^2 in powers of x where chi(), f() are Ramanujan theta functions.
Expansion of q^(-5/6) * (eta(q) * eta(q^4) * eta(q^6)^4 / (eta(q^2)^2 * eta(q^3) * eta(q^12)))^2 in powers of q.
Euler transform of period 12 sequence [-2, 2, 0, 0, -2, -4, -2, 0, 0, 2, -2, -4, ...].
a(n) = (-1)^n * A098098(n) = A208435(2*n + 1) = A208457(2*n + 1). 6 * a(n) = A121613(3*n + 2).
Convolution square of A258832.

A207541 Expansion of phi(q)^3 * phi(-q) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 4, 0, -16, -8, 24, 0, -32, 24, 52, 0, -48, -32, 56, 0, -96, 24, 72, 0, -80, -48, 128, 0, -96, 96, 124, 0, -160, -64, 120, 0, -128, 24, 192, 0, -192, -104, 152, 0, -224, 144, 168, 0, -176, -96, 312, 0, -192, 96, 228, 0, -288, -112, 216, 0, -288, 192, 320, 0
Offset: 0

Views

Author

Michael Somos, Feb 26 2012

Keywords

Comments

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

Examples

			1 + 4*q - 16*q^3 - 8*q^4 + 24*q^5 - 32*q^7 + 24*q^8 + 52*q^9 - 48*q^11 + ...

Links

Crossrefs

Cf. A096727, A097723, A112610, A121613, A139093, A208435.

Programs

  • Mathematica
    a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]^3*EllipticTheta[3, 0, -q], {q, 0, n}]; Table[A207541[n], {n, 0, 50}] (* G. C. Greubel, Dec 16 2017 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^7 / (eta(x + A)^2 * eta(x^4 + A)^3))^2, n))}

Formula

Expansion of phi(-q^4)^4 + 4 * q * psi(-q^2)^4 = phi(q)^3 * phi(-q) = phi(q)^2 * phi(-q^2)^2 = psi(q)^4 * chi(-q^2)^6 = phi(-q^2)^6 / phi(-q)^2 = f(q)^6 / psi(q)^2 = f(q)^4 * chi(-q^2)^2 in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (eta(q^2)^7 / (eta(q)^2 * eta(q^4)^3))^2 in powers of q.
Euler transform of period 4 sequence [ 4, -10, 4, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 128 (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is g.f. for A112610.
G.f.: Product_{k>0} (1 - x^(2*k))^14 / ((1 - x^k)^4 * (1 - x^(4*k))^6).
a(3*n + 2) = 24 * A208435(n). a(4*n + 2) = 0. a(2*n + 1) = 4 * A121613(n). a(4*n) = A096727(n). a(4*n + 1) = 4 * A112610(n). a(4*n + 3) = -16 * A097723(n). Convolution square of A139093.

A208457 Expansion of x * f(-x) * f(-x^12)^3 * psi(-x^3) / psi(x^2) in powers of x where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

0, 1, -1, -2, 0, 3, 2, -4, 0, 5, -1, -8, 0, 7, 4, -8, 0, 9, -8, -10, 0, 14, 6, -12, 0, 16, -6, -14, 0, 15, 8, -20, 0, 17, -14, -18, 0, 19, 10, -24, 0, 26, -1, -22, 0, 23, 16, -28, 0, 25, -20, -32, 0, 32, 14, -28, 0, 29, -12, -30, 0, 38, 16, -32, 0, 33, -31
Offset: 0

Views

Author

Michael Somos, Feb 26 2012

Keywords

Comments

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

Examples

			G.f. = x - x^2 - 2*x^3 + 3*x^5 + 2*x^6 - 4*x^7 + 5*x^9 - x^10 - 8*x^11 + ...
G.f. = q^5 - q^8 - 2*q^11 + 3*q^17 + 2*q^20 - 4*q^23 + 5*q^29 - q^32 - 8*q^35 + ...

Links

Crossrefs

Cf. A208435.

Programs

  • Mathematica
    QP:= QPochhammer; Join[{0}, CoefficientList[Series[Simplify[QP[q]* QP[q^2]*QP[q^3]*QP[q^12]^4/(QP[q^4]^2*QP[q^6]), q > 0], {q, 0, 50}], q]] (* G. C. Greubel, Aug 12 2018 *)
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)^4 / (eta(x^4 + A)^2 * eta(x^6 + A)), n))};

Formula

Expansion of q^(-2/3) * eta(q) * eta(q^2) * eta(q^3) * eta(q^12)^4 / (eta(q^4)^2 * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [-1, -2, -2, 0, -1, -2, -1, 0, -2, -2, -1, -4, ...].
a(4*n) = 0. a(n) = -(-1)^n * A208435(n).
Showing 1-3 of 3 results.

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