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.

A122861 Expansion of phi(-q)chi(-q)psi(q^3) in powers of q where phi(),chi(),psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -3, 2, 0, 2, -3, 2, 0, 1, -6, 2, 0, 2, 0, 2, 0, 3, -6, 0, 0, 2, -3, 2, 0, 2, -6, 2, 0, 0, 0, 4, 0, 2, -3, 2, 0, 2, -6, 0, 0, 1, -6, 2, 0, 4, 0, 2, 0, 0, -6, 2, 0, 2, 0, 2, 0, 3, -6, 2, 0, 2, 0, 0, 0, 2, -9, 2, 0, 0, -6, 2, 0, 4, 0, 2, 0, 2, 0, 0, 0, 2, -6, 4, 0, 0, -3, 4, 0, 0, -6, 2, 0, 2, 0, 2, 0, 1, -6, 0, 0, 4, -6, 2, 0, 2
Offset: 0

Views

Author

Michael Somos, Sep 15 2006

Keywords

Comments

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Links

Crossrefs

Cf. A033687, A097109, A097195, A115979, A129576.
Cf. A000122, A000700, A010054, A121373.

Programs

  • Mathematica
    A122861[n_] := SeriesCoefficient[(QPochhammer[q]^3*QPochhammer[q^6]^2)/(QPochhammer[q^2]^2 *QPochhammer[q^3]), {q, 0, n}]; Table[A122861[n], {n, 0, 50}] (* G. C. Greubel, Oct 05 2017 *)
  • PARI
    {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^3*eta(x^6+A)^2/eta(x^2+A)^2/eta(x^3+A), n))}
  • PARI
    {a(n)=local(A, p, e); if(n<0, 0, n=3*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 3*(e%2-1), if(p==3, 0, if(p%6==1, e+1, !(e%2)))))))}

Formula

Expansion of q^(-1/3)*eta(q)^3*eta(q^6)^2/(eta(q^2)^2*eta(q^3)) in powers of q.
Euler transform of period 6 sequence [ -3, -1, -2, -1, -3, -2, ...].
a(n) = b(3n+1) where b(n) is multiplicative and b(2^e) = -3(1+(-1)^e)/2 if e>0, b(3^e) = 0^e, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
a(4n+3) = 0.
a(n) = A115979(3n+1) = A097109(3n+1).
a(2n) = A097195(n) = A033687(2n); a(2n+1) = -3*A033687(2n+1).
a(n) = (-1)^n * A129576(n). - Amiram Eldar, Jan 28 2024

A123530 Expansion of q^(-1/2)*eta(q)^2*eta(q^6)^3/(eta(q^2)*eta(q^3)^2) in powers of q.

Original entry on oeis.org

1, -2, 0, 2, -2, 0, 2, 0, 0, 2, -4, 0, 1, -2, 0, 2, 0, 0, 2, -4, 0, 2, 0, 0, 3, 0, 0, 0, -4, 0, 2, -4, 0, 2, 0, 0, 2, -2, 0, 2, -2, 0, 0, 0, 0, 4, -4, 0, 2, 0, 0, 2, 0, 0, 2, -4, 0, 0, -4, 0, 1, 0, 0, 2, -4, 0, 4, 0, 0, 2, 0, 0, 0, -6, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 3, -4, 0, 2, 0, 0, 2, -4, 0, 0, -4, 0, 2, 0, 0, 2, -4, 0, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Oct 02 2006

Keywords

Links

Programs

  • Mathematica
    QP = QPochhammer; s = QP[q]^2*(QP[q^6]^3/(QP[q^2]*QP[q^3]^2)) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
  • PARI
    {a(n)=if(n<0, 0, n=2*n+1; sumdiv(n, d, kronecker(-12,d)*[0,1,0,-2,0,1][n/d%6+1]))}
  • PARI
    {a(n)=local(A, p, e); if(n<0, 0, n=2*n+1; A=factor(n); prod( k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 0, if(p==3, -2, if(p%6==1, e+1, !(e%2)))))))}
  • PARI
    {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^2*eta(x^6+A)^3/eta(x^2+A)/eta(x^3+A)^2, n))}

Formula

Euler transform of period 6 sequence [ -2, -1, 0, -1, -2, -2, ...].
a(n) = b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = -2 if e>0, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
G.f.: Sum_{k>0} F(x^(6k-5))-F(x^(6k-1)) where F(x)=(x-x^3)/(1+x^2+x^4).
a(3*n+2) = 0.
a(3*n) = A097195(n).
a(3*n+1) = -2*A033762(n).
a(n) = A097109(2*n+1) = A112848(2*n+1).

A253243 Expansion of phi(-x^2) * psi(x^3) * chi(x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Views

Author

Michael Somos, Jun 04 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^2 + 2*x^3 - 4*x^5 + x^6 + 2*x^9 + 3*x^12 - 4*x^14 + 2*x^15 + ...
G.f. = q - 2*q^9 + 2*q^13 - 4*q^21 + q^25 + 2*q^37 + 3*q^49 - 4*q^57 + ...

Links

Crossrefs

Cf. A005928, A097109, A112605, A112848, A123477, A123530, A123884, A226535, A246650.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2] QPochhammer[ -x^3, x^6] EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8)), {x, 0, n}];
  • PARI
    {a(n) = if( n<0, 0, n = 4*n + 1; sumdiv(n, d, [ 0, 1, -1, -3, 1, -1, 3, 1, -1] [d%9 + 1]))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A)^4 / (eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^12 + A)), n))};

Formula

Expansion q^(-1/4) * eta(q^2)^2 * eta(q^6)^4 / (eta(q^3)^2 * eta(q^4) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 0, -2, 2, -1, 0, -4, 0, -1, 2, -2, 0, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 108^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246650.
a(n) = A123530(2*n) = A097109(4*n + 1) = A112848(4*n + 1) = A123477(4*n + 1). 3 * a(n) = A226535(4*n + 1). -3 * a(n) = A005928(4*n + 1).
a(3*n) = A123884(n). a(3*n + 1) = 0. a(3*n + 2) = -2 * A112605(n).
Showing 1-3 of 3 results.

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