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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A262726 Expansion of phi(-x) * psi(x^6) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Views

Author

Michael Somos, Sep 28 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*x^4 + x^6 - 2*x^7 - 2*x^9 + 2*x^10 - 2*x^15 + 2*x^16 + ...
G.f. = q^3 - 2*q^7 + 2*q^19 + q^27 - 2*q^31 - 2*q^39 + 2*q^43 - 2*q^63 + ...

Links

Crossrefs

Cf. A112605, A112608, A112609, A138270, A262780.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 4 n + 3, KroneckerSymbol[ -3, #] &]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^3] / (2 x^(3/4)), {x, 0, n}];
a[ n_] := If[ n < 0, 0, Times @@ (Which[ # < 5, Mod[#, 2], Mod[#, 6] == 5, 1 - Mod[#2, 2], True, (#2  + 1) KroneckerSymbol[ 6, #]^#2] & @@@ FactorInteger @ (4 n + 3))]; (* Michael Somos, Oct 01 2015 *)
  • PARI
    {a(n) = if( n<0, 0, (-1)^n * sumdiv(4*n + 3, d, kronecker(-3, d)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^12 + A)^2 / (eta(x^2 + A) * eta(x^6 + A)), n))};
  • PARI
    {a(n) = my(A, p, e); if( n<0, 0, A = factor(4*n + 3); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, p%2, p%6 == 1, (e+1) * if( p%24 == 1 || p%24 == 19, 1, (-1)^e), 1-e%2 )))}; /* Michael Somos, Oct 01 2015 */

Formula

Expansion of q^(-3/4) * eta(q)^2 * eta(q^12)^2 / (eta(q^2) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [-2, -1, -2, -1, -2, 0, -2, -1, -2, -1, -2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = 192^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(n) = (-1)^n * A112605(n). -2 * a(n) = A138270(2*n + 1).
a(2*n) = A112608(n). a(2*n + 1) = -2 * A112609(n). a(3*n + 2) = 0.
a(n) = A262780(2*n + 1). - Michael Somos, Oct 01 2015

A193514 Expansion of phi(-q)^2 * phi(-q^9) / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Views

Author

Michael Somos, Jul 29 2011

Keywords

Comments

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

Examples

			G.f. = 1 - 4*q + 4*q^2 + 2*q^3 - 4*q^4 + 4*q^6 - 8*q^7 + 4*q^8 + 2*q^9 + 2*q^12 + ...

Links

Crossrefs

Cf. A033687, A033762, A097195, A112604, A112605.

Programs

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

Formula

Expansion of (-2 * a(q) +2 * a(q^2) +3 * a(q^3)) / 3 = b(q) * (b(q) + 2 * b(q^2)) / (3 * b(q^2)) in powers of q where a(), b() are cubic AGM functions.
Expansion of eta(q)^4 * eta(q^6) * eta(q^9)^2 / (eta(q^2)^2 * eta(q^3)^2 * eta(q^18)) in powers of q.
Euler transform of period 18 sequence [ -4, -2, -2, -2, -4, -1, -4, -2, -4, -2, -4, -1, -4, -2, -2, -2, -4, -2, ...].
Moebius transform is period 18 sequence [ -4, 8, 6, -8, 4, -6, -4, 8, 0, -8, 4, 6, -4, 8, -6, -8, 4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 432^(1/2) (t / i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A193426.
a(3*n) = A123330(n). a(3*n + 1) = -4 * A033687(n). a(6*n + 1) = -4 * A097195(n). a(6*n + 2) = 4 * A033687(n). a(6*n + 3) = 2 * A033762(n). a(6*n + 4) = 4 * A033687(n). a(8*n + 2) = 4 * A112604(n). a(8*n + 6) = 4 * A112605(n). a(6*n + 5) = 0. a(4*n) = a(n).

A253623 Expansion of phi(q) * f(q, q^2)^2 / f(q^2, q^4) in powers of q where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 4, 6, 4, 0, 0, 6, 8, 6, 4, 0, 0, 0, 8, 12, 0, 0, 0, 6, 8, 0, 8, 0, 0, 6, 4, 12, 4, 0, 0, 0, 8, 6, 0, 0, 0, 0, 8, 12, 8, 0, 0, 12, 8, 0, 0, 0, 0, 0, 12, 6, 0, 0, 0, 6, 0, 12, 8, 0, 0, 0, 8, 12, 8, 0, 0, 0, 8, 0, 0, 0, 0, 6, 8, 12, 4, 0, 0, 12, 8, 0, 4, 0, 0
Offset: 0

Views

Author

Michael Somos, Jan 06 2015

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = 1 + 4*x + 6*x^2 + 4*x^3 + 6*x^6 + 8*x^7 + 6*x^8 + 4*x^9 + 8*x^13 + ...

Links

Crossrefs

Cf. A004016, A033687, A033762, A097195, A112604, A112605, A121361, A123884, A186706, A244339, A253625.
Cf. A000122, A000700, A005882, A005928, A010054, A121373.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(12), 1), 83); A[1] + 4*A[2] + 6*A[3] + 4*A[4];
  • Mathematica
    a[ n_] := SeriesCoefficient[ 1 + 2 Sum[ (1 + Mod[k, 2]) q^k / (1 - q^k + q^(2 k)), {k, n}], {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^2 EllipticTheta[ 4, 0, q^3]^2 / (EllipticTheta[ 4, 0, q^2] EllipticTheta[ 4, 0, q^6]), {q, 0, n}];
a[ n_] := If[ n < 1, Boole[n == 0], 2 (-1)^n Sum[(-1)^(n/d) {2, -1, 0, 1, -2, 0}[[ Mod[ d, 6, 1] ]], {d, Divisors @ n}]];
  • PARI
    {a(n) = if( n<1, n==0, 2 * sumdiv(n, d, (n/d%2 + 1) * (-1)^(d\3) * (d%3>0) ))};
  • PARI
    {a(n) = if( n<1, n==0, 2 * (-1)^n * sumdiv(n, d, (-1)^(n/d) * [0, 2, -1, 0, 1, -2][d%6 + 1]))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 * eta(x^3 + A)^4 * eta(x^12 + A) / (eta(x + A)^4 * eta(x^4 + A)^3 * eta(x^6 + A)^4), n))};
  • PARI
    {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 4 * prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 3/2*(e%2), if( p==3, 1, if( p%6 == 1, e+1, 1-e%2))))))};

Formula

Expansion of phi(q)^2 * phi(-q^3)^2 / (phi(-q^2) * phi(-q^6)) = psi(q) * psi(-q^3) * (chi(q) * chi(-q^3))^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of (2*a(q) + 3*a(q^2) - 2*a(q^4)) / 3 = (b(q) - 2*b(q^4)) * (b(q) - 4*b(q^4)) / (3*b(q^2)) in powers of q where a(), b() are cubic AGM theta functions.
Expansion of eta(q^2)^8 * eta(q^3)^4 * eta(q^12) / (eta(q)^4 * eta(q^4)^3 * eta(q^6)^4) in powers of q.
Euler transform of period 12 sequence [ 4, -4, 0, -1, 4, -4, 4, -1, 0, -4, 4, -2, ...].
Moebius transform is period 12 sequence [ 4, 2, 0, -6, -4, 0, 4, 6, 0, -2, -4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 48^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A253625.
a(n) = 4*b(n) where b() is multiplicative with b(2^e) = (3/4) * (1 - (-1)^e) if e>0, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
G.f.: 1 + Sum_{k>0} (1 + (k mod 2)) * q^k / (1 - q^k + q^(2*k)).
G.f.: Product_{k>0} (1 + q^k) * (1 - q^(2*k)) * (1 - q^(3*k)) * (1 + q^(6*k)) / ((1 + q^(2*k)) * (1 - q^k + q^(2*k)))^3.
a(n) = (-1)^n * A244339(n). a(2*n) = A004016(n). a(2*n + 1) = 4 * A033762(n). a(3*n) = a(n). a(6*n + 1) = 4 * A097195(n). a(6*n + 2) = 6 * A033687(n). a(6*n + 4) = a(6*n = 5) = 0.
a(12*n + 1) = 4 * A123884(n). a(12*n + 2) = 6 * A097195(n). a(12*n + 3) = 4 * A112604(n). a(12*n + 7) = 8 * A121361(n). a(12*n + 9) = 4 * A112605(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(3) = 3.627598... (A186706). - Amiram Eldar, Dec 30 2023
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