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.

A111932 Expansion of q * (psi(q) * psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 1, 4, 6, 2, 8, 8, 1, 12, 12, 4, 14, 16, 6, 16, 18, 2, 20, 24, 8, 24, 24, 8, 31, 28, 1, 32, 30, 12, 32, 32, 12, 36, 48, 4, 38, 40, 14, 48, 42, 16, 44, 48, 6, 48, 48, 16, 57, 62, 18, 56, 54, 2, 72, 64, 20, 60, 60, 24, 62, 64, 8, 64, 84, 24, 68, 72, 24, 96, 72, 8, 74, 76, 31
Offset: 1

Views

Author

Michael Somos, Aug 21 2005, Apr 18 2007

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. = q + 2*q^2 + q^3 + 4*q^4 + 6*q^5 + 2*q^6 + 8*q^7 + 8*q^8 + q^9 + ...

References

  • Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 223 Entry 3(iii).
  • Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 87, Eq. (33.2).

Links

Crossrefs

Cf. A033762, A131946, A222171.
Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

Programs

  • Mathematica
    a[ n_] := If[ n < 1, 0, Sum[ Mod[n/d, 2] d KroneckerSymbol[ 9, d], { d, Divisors[ n]}]]; (* Michael Somos, Sep 19 2013 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^2] QPochhammer[ q^6])^4 / (QPochhammer[ q] QPochhammer[ q^3])^2, {q, 0, n}]; (* Michael Somos, Sep 19 2013 *)
  • PARI
    {a(n) = if( n<1, 0, sumdiv(n, d, (n/d % 2) * d * (d%3>0)))};
  • PARI
    {a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p==2, p^e, if( p==3, 1, (p^(e+1) - 1) / (p-1)))))) };
  • PARI
    {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^4 / (eta(x + A) * eta(x^3 + A))^2, n))};
  • Sage
    A = ModularForms( Gamma0(6), 2, prec=50) . basis();  A[1] + 2*A[2]; # Michael Somos, Sep 19 2013

Formula

Expansion of (1/3) * (b(q^2)^2 / b(q))* (c(q^2)^2 / c(q)) in powers of q where b(), c() are cubic AGM theta functions.
Expansion of (eta(q^2) * eta(q^6))^4 / (eta(q) * eta(q^3))^2 in powers of q.
Euler transform of period 6 sequence [ 2, -2, 4, -2, 2, -4, ...].
Multiplicative with a(2^e) = 2^e, a(3^e) = 1, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u*w * (u - 4*v) - v * (v - 4*w)^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (3/4) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A131946. - Michael Somos, Sep 19 2013
G.f.: Sum_{k>0} k * x^k * (1 - x^(2*k))^2 / (1 - x^(6*k)) = x * Product_{k>0} ((1 + x^k) * (1 + x^(3*k)))^4 * ((1 - x^k) * (1 - x^(3*k)))^2.
a(3*n) = a(n), a(2*n) = 2 * a(n).
Convolution square of A033762. - Michael Somos, Sep 19 2013
From Amiram Eldar, Sep 12 2023: (Start)
Dirichlet g.f.: (1 - 1/2^s) * (1 - 1/3^(s-1)) * zeta(s-1) * zeta(s).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/24 = 0.411233... (A222171). (End)

A186100 Expansion of 2 * a(q^2)^2 - a(q)^2 in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

1, -12, -12, -12, -12, -72, -12, -96, -12, -12, -72, -144, -12, -168, -96, -72, -12, -216, -12, -240, -72, -96, -144, -288, -12, -372, -168, -12, -96, -360, -72, -384, -12, -144, -216, -576, -12, -456, -240, -168, -72, -504, -96, -528, -144, -72, -288
Offset: 0

Views

Author

Michael Somos, Feb 12 2011

Keywords

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

Examples

			G.f. = 1 - 12*q - 12*q^2 - 12*q^3 - 12*q^4 - 72*q^5 - 12*q^6 - 96*q^7 + ...

Links

Crossrefs

Cf. A008653, A098098, A111932, A121443, A131943, A131946, A186099.

Programs

  • Mathematica
    a[ n_] := If[ n < 1, Boole[n == 0], -12 DivisorSum[ n, # Boole[ 1 == GCD[#, 6]] &]]; (* Michael Somos, Jul 07 2015 *)
a[ n_] := SeriesCoefficient[(EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^3])^2 - 1/2 (EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)])^2, {x, 0, n}]; (* Michael Somos, Jul 07 2015 *)
  • PARI
    {a(n) = if( n<1, n==0, -12 * sumdiv( n, d, d * (1 == gcd( d, 6) ) ) )};
  • PARI
    {a(n) = if( n<1, n==0, -12 * direuler( p=2, n, 1 / (1 - X) / (1 - (p>3) * p * X)) [n])};

Formula

Expansion of b(x) * b(x^2) - c(x) * c(x^2) in powers of x where b(), c() are cubic AGM functions.
Expansion of (phi(-x) * phi(-x^3))^2 - 8 * x * (psi(x) * psi(x^3))^2 in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of (P(q) - 2*P(q^2) - 3*P(q^3) + 6*P(q^6)) / 2 in powers of q where P() is a Ramanujan Eisenstein series. - Michael Somos, Jul 07 2015
a(n) = -12 * A186099(n) if n>0. a(2*n) = a(n). a(2*n + 1) = - A008653(2*n + 1). a(n) = 2 * A008653(n) - A008653(2*n) = A131946(n) - 8 * A111932(n) = A131943(n) - 9 * A121443(n).
a(3*n) = a(n). a(6*n + 5) = -72 * A098098(n).- Michael Somos, Jul 07 2015

A131947 Expansion of (1 - (phi(-q) * phi(-q^3))^2)/4 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 1, -5, 6, -1, 8, -13, 1, -6, 12, -5, 14, -8, 6, -29, 18, -1, 20, -30, 8, -12, 24, -13, 31, -14, 1, -40, 30, -6, 32, -61, 12, -18, 48, -5, 38, -20, 14, -78, 42, -8, 44, -60, 6, -24, 48, -29, 57, -31, 18, -70, 54, -1, 72, -104, 20, -30, 60, -30, 62
Offset: 1

Views

Author

Michael Somos, Jul 30 2007

Keywords

Comments

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

Examples

			G.f. = x - x^2 + x^3 - 5*x^4 + 6*x^5 - x^6 + 8*x^7 - 13*x^8 + x^9 - 6*x^10 + ...

References

  • Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.66).

Links

Crossrefs

Cf. A113262, A131946.
Cf. A000122, A000700, A010054, A121373.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (1 - (EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^3])^2) / 4, {q, 0, n}]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := SeriesCoefficient[ (1 - (QPochhammer[ q] QPochhammer[ q^3])^4 / (QPochhammer[ q^2] QPochhammer[ q^6])^2) / 4, {q, 0, n}]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := If[ n < 1, 0, Sum[ d {0, 1, -1, 0, -1, 1}[[Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := If[ n < 1, 0, Sum[ n/d {6, 1, -3, -2, -3, 1}[[Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* Michael Somos, Nov 11 2015 *)
  • PARI
    {a(n) = if( n<1, 0, sumdiv(n, d, d*((abs(d%6-3) == 2) - (abs(d%6-3) == 1))))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - (eta(x + A) * eta(x^3 + A))^4 / (eta(x^2 + A) * eta(x^6 + A))^2) / 4, n))};
  • 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, 3 - p^(e+1), p==3, 1, (p^(e+1) - 1) / (p-1) )))};

Formula

a(n) is multiplicative with a(2^e) = 3 - 2^(e+1), a(3^e) = 1, a(p^e) = (p^(e+1) - 1) / (p-1) if p>3.
G.f.: Sum_{k>0} k * (-x)^k / (1 - x^k) * Kronecker(9, k) = ((theta_3(-x) * theta_3(-x^3))^2 - 1) / 4.
a(n) = -(-1)^n * A113262(n). -4 * a(n) = A131946(n) unless n=0.
Dirichlet g.f.: (1 - 1/2^(s-2)) * (1 - 1/3^(s-1)) * zeta(s-1) * zeta(s). - Amiram Eldar, Sep 12 2023
Showing 1-3 of 3 results.

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