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 1-2 of 2 results.

A122190 Expansion of q^(-1/4) * eta(q^2) * eta(q^5)^3 / (eta(q) * eta(q^10)) in powers of q.

Original entry on oeis.org

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

Views

Author

Michael Somos, Aug 24 2006

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution product of A133100 and A133101. - Michael Somos, Feb 10 2015

Examples

			G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + x^6 + 2*x^7 + 2*x^9 + 2*x^10 + x^11 + ...
G.f. = q + q^5 + q^9 + 2*q^13 + 2*q^17 + q^25 + 2*q^29 + 2*q^37 + 2*q^41 + ...

Links

Crossrefs

Cf. A019694, A053694, A094247, A133100, A133101, A133573.
Cf. A000122, A000700, A010054, A121373.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^5]^3 / (QPochhammer[ x] QPochhammer[ x^10]), {x, 0, n}]; (* Michael Somos, Feb 10 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^5 + A)^3 / (eta(x + A) * eta(x^10 + A)), n))};
  • PARI
    {a(n) = my(A, p, e); if( n<0, 0, n = 4*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==5, 1, if( p%4==1, e+1, (1 + (-1)^e) / 2))))))};

Formula

Euler transform of period 10 sequence [ 1, 0, 1, 0, -2, 0, 1, 0, 1, -2, ...].
a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(5^e) = 1, b(p^e) = e+1 if p == 1 (mod 4), b(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).
G.f.: Product_{k>0} (1 - x^(5*k))^2 * (1 + x^k) / (1 + x^(5*k)).
G.f.: Sum_{k>=0} a(k) * x^(4k+1) = Sum_{k>0 odd} x^k * (1 - x^(2*k)) * (1 - x^(6*k)) / (1 + x^(10*k)).
Expansion of f(x, x^4) * f(x^2, x^3) in powers of x where f() is the Ramanujan two-variable theta function.
Expansion of psi(x)^2 - x * psi(x^5)^2 in powers of x where psi() is a Ramanujan theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (40 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133573.
a(n) = A053694(4*n) = A094247(4*n + 1).
a(3*n + 2) = a(5*n + 1) = a(n). - Michael Somos, Feb 10 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/5 = 1.256637... (A019694). - Amiram Eldar, Dec 29 2023

A133574 Expansion of (5 * phi(q^5)^2 - phi(q)^2) / 4 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, -1, 0, -1, 3, 0, 0, -1, -1, 3, 0, 0, -2, 0, 0, -1, -2, -1, 0, 3, 0, 0, 0, 0, 7, -2, 0, 0, -2, 0, 0, -1, 0, -2, 0, -1, -2, 0, 0, 3, -2, 0, 0, 0, 3, 0, 0, 0, -1, 7, 0, -2, -2, 0, 0, 0, 0, -2, 0, 0, -2, 0, 0, -1, 6, 0, 0, -2, 0, 0, 0, -1, -2, -2, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Sep 17 2007

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 + 3*q^5 - q^8 - q^9 + 3*q^10 - 2*q^13 - q^16 + ...

Links

Crossrefs

Cf. A053694, A133573.
Cf. A000700, A000122, A010054, A121373.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(20), 1), 78); A[1] - A[2] - A[3] - A[5] + 3*A[6] - A[9]; /* Michael Somos, Oct 31 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ (5 EllipticTheta[ 3, 0, q^5]^2 - EllipticTheta[ 3, 0, q]^2)/4, {q, 0, n}]; (* Michael Somos, Oct 31 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^2 QPochhammer[ -q^5, q^10] / QPochhammer[ -q, q^2], {q, 0, n}]; (* Michael Somos, Oct 31 2015 *)
a[ n_] := SeriesCoefficient[ (1/2) x^(-1/4) EllipticTheta[ 2, Pi/4, x^(1/2)]^2 QPochhammer[ -x, x^2] QPochhammer[ -x^5, x^10], {x, 0, n}]; (* Michael Somos, Oct 31 2015 *)
a[ n_] := If[n < 1, Boole[n == 0], DivisorSum[ n, If[Mod[#, 5] == 0, 5 KroneckerSymbol[-4, #/5], 0] - KroneckerSymbol[-4, #] &]]; (* Michael Somos, Oct 31 2015 *)
  • PARI
    {a(n) = if( n<1, n==0, sumdiv(n, d, if( d%5==0, kronecker( -4, d/5) * 5) - kronecker( -4, d)))};
  • PARI
    {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); - prod(k = 1, matsize(A) [1], [p, e] = A[k, ]; if(p == 2, 1, p == 5, 1 - 4*e, p%4 == 1, e+1, 1-e%2 )))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^10 + A)^2 / (eta(x^5 + A) * eta(x^20 + A)), n))};

Formula

Expansion of psi(-q)^2 * chi(q) * chi(q^5) in powers of q where psi(), chi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^4) * eta(q^10)^2 / (eta(q^5) * eta(q^20)) in powers of q.
Euler transform of period 20 sequence [ -1, -1, -1, -2, 0, -1, -1, -2, -1, -2, -1, -2, -1, -1, 0, -2, -1, -1, -1, -2, ...].
Moebius transform is period 20 sequence [ -1, 0, 1, 0, 4, 0, 1, 0, -1, 0, 1, 0, -1, 0, -4, 0, -1, 0, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4), A(x^8)) where f(u1, u2, u4, u8) = (u1 - u2)^2 * (u4 - 2*u8)^2 - u2 * u4 * (u2 - u4) * (u2 - 2*u4).
a(n) = -b(n) where b() is multiplicative with b(2^e) = 1, b(5^e) = 1-4*e, b(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 10 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A053694.
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(4*k)) * (1 + x^(5*k)) / (1 + x^(10*k)).
G.f.: 1 - (Sum_{k>0} x^k / (1 + x^(2*k)) - 5 * x^(5*k) / (1 + x^(10*k))).
a(n) = (-1)^n * A133573(n).
Sum_{k=1..n} abs(a(k)) ~ (8*Pi/25) * n. - Amiram Eldar, Jan 27 2024
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