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.

A034896 Number of solutions to a^2 + b^2 + 3*c^2 + 3*d^2 = n.

Original entry on oeis.org

1, 4, 4, 4, 20, 24, 4, 32, 52, 4, 24, 48, 20, 56, 32, 24, 116, 72, 4, 80, 120, 32, 48, 96, 52, 124, 56, 4, 160, 120, 24, 128, 244, 48, 72, 192, 20, 152, 80, 56, 312, 168, 32, 176, 240, 24, 96, 192, 116, 228, 124, 72, 280, 216, 4, 288, 416, 80, 120, 240, 120, 248, 128, 32, 500
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Number 16 of the 126 eta-quotients listed in Table 1 of Williams 2012. - Michael Somos, Nov 10 2018

Examples

			G.f. = 1 + 4*x + 4*x^2 + 4*x^3 + 20*x^4 + 24*x^5 + 4*x^6 + 32*x^7 + ... - _Michael Somos_, Nov 10 2018

References

  • B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 223, Entry 3(iv).
  • L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 229.
  • N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.3), p. 76, Eq. (31.43).

Links

Crossrefs

Cf. A272364, A320147, A320148.
Cf. A033716, A113262, A134946.

Programs

  • Mathematica
    A034896[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^3])^2, {q, 0, n}]; Table[A034896[n], {n, 0, 50}] (* G. C. Greubel, Dec 24 2017 *)
a[ n_] := If[ n < 1, Boole[n == 0], 4 DivisorSum[ n, # KroneckerSymbol[ 9, #] (-1)^(n + #) &]]; (* Michael Somos, Nov 10 2018 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^10 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A))^4, n))}; /* Michael Somos, Nov 10 2018 */

Formula

Expansion of theta_3(q)^2*theta_3(q^3)^2.
G.f.: s(2)^10*s(6)^10/(s(1)*s(3)*s(4)*s(12))^4, where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind's function, cf. A010815. [Fine]
Fine gives an explicit formula for a(n) in terms of the divisors of n.
From Michael Somos, Nov 10 2018: (Start)
Expansion of (a(q) + 2*a(q^4))^2 / 9 = (a(q)^2 - 2*a(q^2)^2 + 4*a(q^4)^2) / 3 in powers of q where a() is a cubic AGM theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: 1 + 4 Sum_{k>0} k x^k / (1 - (-x)^k) Kronecker(9, k).
a(n) = 1 + 4 * A113262(n) = (-1)^n * A134946(n). Convolution square of A033716.
a(n) = 4 * (s(n) - 2*s(n/2) - 3*s(n/3) + 4*s(n/4) + 6*s(n/6) - 12*s(n/12)) if n>0 where s(x) = sum of divisors of x for integer x else 0. (End)

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

A132001 Expansion of 1 - (1/3) * b(q) * b(q^2) * c(q)^2 / c(q^2) in powers of q where b(), c() are cubic AGM functions.

Original entry on oeis.org

1, 5, 1, -11, -24, 5, 50, 53, 1, -120, -120, -11, 170, 250, -24, -203, -288, 5, 362, 264, 50, -600, -528, 53, 601, 850, 1, -550, -840, -120, 962, 821, -120, -1440, -1200, -11, 1370, 1810, 170, -1272, -1680, 250, 1850, 1320, -24, -2640, -2208, -203, 2451, 3005
Offset: 1

Views

Author

Michael Somos, Aug 06 2007

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).

Examples

			G.f. = q + 5*q^2 + q^3 - 11*q^4 - 24*q^5 + 5*q^6 + 50*q^7 + 53*q^8 + q^9 - 120*q^10 + ...

References

  • N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.71).

Links

Crossrefs

Cf. A132000, A113262.

Programs

  • Mathematica
    a[ n_] := If[ n < 1, 0, -DivisorSum[ n, #^2 (-1)^# KroneckerSymbol[ -3, #] &]]; (* Michael Somos, Nov 02 2015 *)
a[ n_] := SeriesCoefficient[ 1 - QPochhammer[ q] QPochhammer[ q^2]^4 QPochhammer[ q^3]^5 / QPochhammer[ q^6]^4, {q, 0, n}]; (* Michael Somos, Nov 02 2015 *)
a[ n_] := SeriesCoefficient[ 1 - (1/4) EllipticTheta[ 4, 0, q]^2 EllipticTheta[ 4, 0, q^3]^2 EllipticTheta[ 2, 0, q^(1/2)]^3 / EllipticTheta[ 2, 0, q^(3/2)], {q, 0, n}]; (* Michael Somos, Nov 02 2015 *)
a[ n_] := SeriesCoefficient[ 1 - (9 EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^3]^5 - EllipticTheta[ 4, 0, q]^5 EllipticTheta[ 4, 0, q^3]) / 8, {q, 0, n}]; (* Michael Somos, Nov 02 2015 *)
  • PARI
    {a(n) = if(n<1, 0, -sumdiv(n, d, d^2 * (-1)^d * kronecker(-3, d)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 - eta(x + A) * eta(x^2 + A)^4 * eta(x^3 + A)^5 / eta(x^6 + A)^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==3, 1, p==2, 2 + ((-4)^(e+1) - 1) / 5, p = p^2 * kronecker(-3, p); (p^(e+1) - 1) / (p-1) )))};
  • PARI
    q='q+O('q^99); Vec(-eta(q)*eta(q^2)^4*eta(q^3)^5/eta(q^6)^4+1) \\ Altug Alkan, Sep 07 2018

Formula

Expansion of 1 - phi(-q)^2 * phi(-q^3)^2 * psi(q)^3 / psi(q^3) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of 1 - eta(q) * eta(q^2)^4 * eta(q^3)^5 / eta(q^6)^4 in powers of q.
a(n) is multiplicative with a(2^e) = 2 + ((-4)^(e+1) - 1)/5, a(3^e) = 1, a(p^e) = (q^(e+1) - 1) / (q - 1) where q = p^2 * Kronecker(-3, p) if p > 3.
a(3*n) = a(n).
G.f.: Sum_{k>0} k^2 * Kronecker(-3,k) * x^k / (1 - (-x)^k) = 1 - Product_{k>0} (1 - x^(3k)) * (1 - x^k)^5 / (1 - x^k + x^(2k))^4.
a(n) = - A132000(n) unless n = 0.
Expansion of 1 - (9 * phi(-q) * phi(-q^3)^5 - phi(-q)^5 * phi(-q^3)) / 8 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Nov 02 2015
a(n) = -(-1)^n * A113262(n) unless n = 0. - Michael Somos, Nov 02 2015
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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