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.

A113417 Expansion of phi(x) * phi(-x)^2 * psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, -4, 8, 7, -10, -12, 8, 18, -18, -16, 24, 21, -20, -28, 32, 20, -32, -36, 24, 42, -42, -28, 48, 57, -36, -52, 40, 36, -58, -60, 56, 48, -66, -48, 72, 74, -42, -80, 80, 61, -82, -72, 56, 90, -96, -64, 72, 98, -70, -100, 104, 64, -106, -108, 72, 114, -96
Offset: 0

Views

Author

Michael Somos, Oct 29 2005

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^2 + 8*x^3 + 7*x^4 - 10*x^5 - 12*x^6 + 8*x^7 + 18*x^8 + ...
G.f. = q - 2*q^3 - 4*q^5 + 8*q^7 + 7*q^9 - 10*q^11 - 12*q^13 + 8*q^15 + ...

Links

Crossrefs

Cf. A113419, A117000, A209940, A258096.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(16), 2), 117); A[2] - 2*A[4] - 4*A[6] + 8*A[8] + 7*A[10] - 10*A[12] - 12*A[14]; /* Michael Somos, May 19 2015 */
  • Mathematica
    a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, KroneckerSymbol[ 2, #] # &]]; (* Michael Somos, May 19 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^8]^2 QPochhammer[ x^2]^3 / QPochhammer[ x^4]^3, {x, 0, n}]; (* Michael Somos, May 19 2015 *)
  • PARI
    {a(n) = if( n<0, 0, sumdiv( 2*n + 1, d, d * (d%2) * (-1)^((d + 1) \ 4)))};
  • PARI
    {a(n) = my(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p *= kronecker( 2, p); (p^(e+1) - 1) / (p - 1))))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^8 + A))^2 * (eta(x^2 + A) / eta(x^4 + A))^3, n))};

Formula

Expansion of q^(-1/2) * (eta(q) * eta(q^8))^2 * (eta(q^2) / eta(q^4))^3 in powers of q.
Euler transform of period 8 sequence [ -2, -5, -2, -2, -2, -5, -2, -4, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 7 (mod 8), b(p^e) = ((-p)^(e+1) - 1) / (-p - 1) if p == 3, 5 (mod 8).
G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} (2*k - 1) * (-1)^[k/2] * x^(2*k - 1) / (1 - x^(4*k - 2)) = x * (Product_{k>0} ((1 - x^(2*k)) * (1 - x^(4*k)) * (1 + x^(8*k)))^2 / (1 + x^(4*k))).
a(n) = (-1)^n * A113419(n) = (-1)^floor(n/2) * A209940(n) = (-1)^(n + floor(n/2)) * A258096(n). - Michael Somos, May 19 2015
a(n) = A117000(2*n + 1).
a(n) = Sum_{d | 2*n + 1} Kronecker(2, d) * d.

A113419 Expansion of phi(x)^2 * phi(-x) * psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, -4, -8, 7, 10, -12, -8, 18, 18, -16, -24, 21, 20, -28, -32, 20, 32, -36, -24, 42, 42, -28, -48, 57, 36, -52, -40, 36, 58, -60, -56, 48, 66, -48, -72, 74, 42, -80, -80, 61, 82, -72, -56, 90, 96, -64, -72, 98, 70, -100, -104, 64, 106, -108, -72, 114, 96
Offset: 0

Views

Author

Michael Somos, Oct 29 2005

Keywords

Comments

Number 46 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + 2*x - 4*x^2 - 8*x^3 + 7*x^4 + 10*x^5 - 12*x^6 - 8*x^7 + 18*x^8 + ...
G.f. = q + 2*q^3 - 4*q^5 - 8*q^7 + 7*q^9 + 10*q^11 - 12*q^13 - 8*q^15 + ...

Links

Crossrefs

Cf. A109039, A113417, A209940, A258096.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^9 QPochhammer[ q^8]^2 / (QPochhammer[ q]^2 QPochhammer[ q^4]^5), {q, 0, n}]; (* Michael Somos, May 19 2015 *)
a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 2 n + 1, KroneckerSymbol[ 2, #] # &]]; (* Michael Somos, May 19 2015 *)
  • PARI
    {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, d*(d%2) * (-1)^((n/d)\2 + (d-1)\4)))};
  • PARI
    {a(n) = my(A, p, e, t); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, t = (-1)^(p\2); p *= kronecker( -2, p); (p^(e+1) - t^(e+1)) / (p - t) )))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^9 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^5), n))};

Formula

Expansion of q^(-1/2) * (eta(q^2)^9 * eta(q^8)^2) / (eta(q)^2 * eta(q^4)^5) in powers of q. - Michael Somos, Mar 14 2012
Euler transform of period 8 sequence [ 2, -7, 2, -2, 2, -7, 2, -4, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (x^(e+1) - y^(e+1)) / (x - y) where x = p * (-1)^floor(p/4) and y = (-1)^floor(p/2).
G.f.: Sum_{k>0} (2*k - 1) * (-1)^[(k - 1)/2] * x^(2*k - 1) / (1 + x^(4*k - 2)).
a(n) = (-1)^n * A113417(n) = (-1)^floor(n/2) * A258096(n) = (-1)^(n + floor(n/2)) * A209940(n).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A109039. - Michael Somos, May 20 2015

A209940 Expansion of psi(x^4) * phi(-x^4)^4 / phi(x) in powers of x where phi(), psi() are Ramanujan theta function.

Original entry on oeis.org

1, -2, 4, -8, 7, -10, 12, -8, 18, -18, 16, -24, 21, -20, 28, -32, 20, -32, 36, -24, 42, -42, 28, -48, 57, -36, 52, -40, 36, -58, 60, -56, 48, -66, 48, -72, 74, -42, 80, -80, 61, -82, 72, -56, 90, -96, 64, -72, 98, -70, 100, -104, 64, -106, 108, -72, 114, -96
Offset: 0

Views

Author

Michael Somos, Mar 16 2012

Keywords

Comments

Number 47 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 - 2*x + 4*x^2 - 8*x^3 + 7*x^4 - 10*x^5 + 12*x^6 - 8*x^7 + 18*x^8 + ...
G.f. = q - 2*q^3 + 4*q^5 - 8*q^7 + 7*q^9 - 10*q^11 + 12*q^13 - 8*q^15 + 18*q^17 + ...

Links

Crossrefs

Cf. A113417, A113419, A258096.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 QPochhammer[ q^4]^9 / (QPochhammer[ q^2]^5 QPochhammer[ q^8]^2), {q, 0, n}]; (* Michael Somos, May 19 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^9 / (eta(x^2 + A)^5 * eta(x^8 + A)^2), n))};
  • PARI
    {a(n) = my(A, p, e, f); if( n<0, 0, A = factor(2*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0^e, p==3, ((-p)^(e+1) - 1) / ((-p) - 1), p *= kronecker( 18, p); (-1)^(e*(p\6)) * (p^(e+1) - 1) / (p - 1))))};

Formula

Expansion of q^(-1/2) * eta(q)^2 * eta(q^4)^9 / (eta(q^2)^5 * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ -2, 3, -2, -6, -2, 3, -2, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 512^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A113419.
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = (1 - (-3)^(e+1)) / 4, b(p^e) = (-1)^(e * [p/6]) * ((p*f)^(e+1) - 1) / (p*f - 1) where f = Kronecker( 18, p).
a(n) = (-1)^n * A258096(n) = (-1)^floor(n/2) * A113419(n) = (-1)^(n + floor(n/2)) * A113417(n).
Showing 1-3 of 3 results.

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