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.

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A256014 Expansion of phi(-q^3)^4 / (phi(-q) * phi(-q^9)) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 4, 0, -2, -8, 0, 0, 4, -4, -4, 0, 0, 4, 0, 0, -2, -8, 4, 0, 8, 0, 0, 0, 0, 6, 8, 0, 0, -8, 0, 0, 4, 0, -4, 0, 4, 4, 0, 0, -4, -8, 0, 0, 0, -8, 0, 0, 0, 2, 12, 0, -4, -8, 0, 0, 0, 0, -4, 0, 0, 4, 0, 0, -2, -16, 0, 0, 8, 0, 0, 0, 4, 4, 8, 0, 0, 0, 0, 0, 8
Offset: 0

Views

Author

Michael Somos, Jun 03 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 + 4*q^2 - 2*q^4 - 8*q^5 + 4*q^8 - 4*q^9 - 4*q^10 + 4*q^13 + ...

Links

Crossrefs

Cf. A002175, A104794, A121444, A122856, A122865, A256280, A258277, A258278.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3]^4 / (EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^9]), {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^8 * eta(x^18 + A) / (eta(x + A)^2 * eta(x^6 + A)^4 * eta(x^9 + A)^2), n))};
  • PARI
    {a(n) = if( n<1, n==0, 2^(n%3) * (-1)^(n\3) * sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};

Formula

Expansion of eta(q^2) * eta(q^3)^8 * eta(q^18) / (eta(q)^2 * eta(q^6)^4 * eta(q^9)^2) in powers of q.
Euler transform of period 18 sequence [ 2, 1, -6, 1, 2, -3, 2, 1, -4, 1, 2, -3, 2, 1, -6, 1, 2, -2, ...].
a(n) = (-1)^n * A256280(n). a(3*n + 1) = 2 * A258277(n). a(3*n + 2) = 4 * A258278(n). a(4*n) = A256280(n). a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0.
a(6*n + 2) = 4 * A122865(n). a(6*n + 4) = -2 * A122856(n). a(9*n) = A104794(n). a(12*n + 1) = A002175(n). a(12*n + 5) = -8 * A121444(n).

A256282 Expansion of f(-q^3) * psi(q^3)^3 / (psi(q) * psi(q^9)) in powers of q where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -1, 1, 0, 1, -2, 0, 0, 1, -4, 2, 0, 0, -2, 0, 0, 1, -2, 4, 0, 2, 0, 0, 0, 0, -3, 2, 0, 0, -2, 0, 0, 1, 0, 2, 0, 4, -2, 0, 0, 2, -2, 0, 0, 0, -8, 0, 0, 0, -1, 3, 0, 2, -2, 0, 0, 0, 0, 2, 0, 0, -2, 0, 0, 1, -4, 0, 0, 2, 0, 0, 0, 4, -2, 2, 0, 0, 0, 0, 0, 2
Offset: 0

Views

Author

Michael Somos, Jun 02 2015

Keywords

Comments

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

Examples

			G.f. = 1 - q + q^2 + q^4 - 2*q^5 + q^8 - 4*q^9 + 2*q^10 - 2*q^13 + q^16 + ...

Links

Crossrefs

Cf. A258256, A258277, A258278.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ q^(1/8) QPochhammer[ q^3] EllipticTheta[ 2, 0, q^(3/2)]^3 / (2 EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 2, 0, q^(9/2)]), {q, 0, n}];
a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, {1, 2, -1, 0}[[Mod[#, 4, 1]]] If[ Divisible[#, 9], 4, 1] (-1)^(Boole[Mod[#, 8] == 6] + #) &]]; (* Michael Somos, Jun 06 2015 *)
a[ n_] := If[ n < 1, Boole[ n==0 ], -Times @@ (Which[ # == 1, 1, # == 2, -1, Mod[#, 4] == 1, #2 + 1, True, If[# == 3, 4, 1] Mod[#2 + 1, 2]] & @@@ FactorInteger[n])]; (* Michael Somos, Jun 06 2015 *)
  • PARI
    {a(n) = if( n<1, n==0, sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + d)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^6 * eta(x^9 + A) / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^18 + A)^2), n))};

Formula

Expansion of eta(q) * eta(q^6)^6 * eta(q^9) / (eta(q^2)^2 * eta(q^3)^2 * eta(q^18)^2) in powers of q.
Euler transform of period 18 sequence [ -1, 1, 1, 1, -1, -3, -1, 1, 0, 1, -1, -3, -1, 1, 1, 1, -1, -2, ...].
Moebius transform is a period 72 sequence.
a(n) = (-1)^n * A258256(n). a(2*n) = A258256(n). a(3*n + 1) = - A258277(n). a(3*n + 2) = A258278(n). a(4*n + 3) = 0.
a(n) = -b(n) where b() is multiplicative with a(0) = 1, b(2^e) = -1 if e>0, b(3^e) = 2 * (1 + (-1)^e), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), a(p^e) = e+1 if p == 1 (mod 4). - Michael Somos, Jun 06 2015

A257900 Expansion of 1/2 - (phi(-q)^2 + phi(-q^9)^2) / 4 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 0, -1, 2, 0, 0, -1, 2, -2, 0, 0, 2, 0, 0, -1, 2, -2, 0, -2, 0, 0, 0, 0, 3, -2, 0, 0, 2, 0, 0, -1, 0, -2, 0, -2, 2, 0, 0, -2, 2, 0, 0, 0, 4, 0, 0, 0, 1, -3, 0, -2, 2, 0, 0, 0, 0, -2, 0, 0, 2, 0, 0, -1, 4, 0, 0, -2, 0, 0, 0, -2, 2, -2, 0, 0, 0
Offset: 1

Views

Author

Michael Somos, May 25 2015

Keywords

Comments

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

Examples

			G.f. = q - q^2 - q^4 + 2*q^5 - q^8 + 2*q^9 - 2*q^10 + 2*q^13 - q^16 + ...

Links

Crossrefs

Cf. A113652, A258277, A258278, A258322.
Cf. A000122, A000700, A010054, A121373.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 1/2 - (EllipticTheta[ 4, 0, q]^2 + EllipticTheta[ 4, 0, q^9]^2) / 4, {q, 0, n}]; (* Michael Somos, Jun 02 2015 *)
  • PARI
    {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, -1, p%4==3, if( p>3, 1, 2) * (1-e%2), e+1)))};

Formula

Expansion of 1/2 - (eta(q)^4 * eta(q^18)^2 + eta(q^2)^2 * eta(q^9)^4) / (2 * eta(q^2) * eta(q^18))^2 in powers of q. - Michael Somos, Jun 02 2015
a(n) is multiplicative with a(2^e) = -1 if e>0, a(3^e) = 1 + (-1)^e if e>0, a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), a(p^e) = e+1 if p == 1 (mod 4).
a(3*n + 1) = A258277(n). a(3*n + 2) = - A258278(n). a(9*n) = 2 * A113652(n). a(9*n + 3) = a(9*n + 6) = 0.
-2 * a(n) = A258322(n) unless n = 0 or n == 2 (mod 3).
Sum_{k=1..n} abs(a(k)) ~ (5*Pi/18) * n. - Amiram Eldar, Jan 29 2024

A256279 Expansion of psi(q) * chi(-q^3) * phi(-q^9) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 0, 0, -1, 0, 0, 0, 0, -4, -2, 0, 0, 2, 0, 0, -1, 0, 4, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 4, 2, 0, 0, -2, 0, 0, 0, 0, -8, 0, 0, 0, 1, 0, 0, -2, 0, 0, 0, 0, 0, -2, 0, 0, 2, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, -4, -2
Offset: 0

Views

Author

Michael Somos, Jun 02 2015

Keywords

Comments

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

Examples

			G.f. = 1 + q - q^4 - 4*q^9 - 2*q^10 + 2*q^13 - q^16 + 4*q^18 + 3*q^25 + ...

Links

Crossrefs

Cf. A002175, A122856, A122865, A256269, A258277.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)] / (2 q^(1/8)) QPochhammer[ q^3, q^6] EllipticTheta[ 4, 0, q^9], {q, 0, n}];
  • PARI
    {a(n) = if( n<1, n==0, (-1)^(n\3) * (n%3<2) * sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^9 + A)^2 / (eta(x + A) * eta(x^6 + A) * eta(x^18 + A)), n))};

Formula

Expansion of eta(q^2)^2 * eta(q^3) * eta(q^9)^2 / (eta(q) * eta(q^6) * eta(q^18)) in powers of q.
Euler transform of period 18 sequence [ 1, -1, 0, -1, 1, -1, 1, -1, -2, -1, 1, -1, 1, -1, 0, -1, 1, -2, ...].
a(n) = (-1)^n * A256269(n). a(4*n) = A256269(n).
a(3*n + 2) = a(4*n + 3) = 0. a(3*n + 1) = A258277(n). a(6*n + 4) = - A122856(n). a(12*n + 1) = A002175(n). a(12*n + 4) = - A122865(n).
Previous Showing 11-14 of 14 results.

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