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.

A132000 Expansion of (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, -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
Offset: 0

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. = 1 - x - 5*x^2 - x^3 + 11*x^4 + 24*x^5 - 5*x^6 - 50*x^7 - 53*x^8 - x^9 + ...

References

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

Links

Crossrefs

Cf. A113261, A122373, A123330, A131943.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(6), 3), 50); A[1] - A[2] - 5*A[3] - A[4]; /* Michael Somos, Nov 03 2015 */
  • Mathematica
    a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, #^2 (-1)^# KroneckerSymbol[ -3, #] &]]; (* Michael Somos, Nov 03 2015 *)
a[ n_] := SeriesCoefficient[ (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 03 2015 *)
a[ n_] := SeriesCoefficient[(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 03 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q] QPochhammer[ q^2]^4 QPochhammer[ q^3]^5 / QPochhammer[ q^6]^4, {q, 0, n}]; (* Michael Somos, Nov 03 2015 *)
  • PARI
    {a(n) = if( n<1, n==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( 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, n==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) )))};

Formula

Expansion of 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 eta(q) * eta(q^2)^4 * eta(q^3)^5 / eta(q^6)^4 in powers of q.
Euler transform of period 6 sequence [-1, -5, -6, -5, -1, -6, ...].
a(n) = -b(n) where b() is multiplicative with b(2^e) = 2+((-4)^(e+1)-1)/5, b(3^e) = 1, b(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.: 1 - Sum_{k>0} k^2 * Kronecker(-3, k) * x^k / (1 - (-x)^k) = Product_{k>0} (1 - x^(3*k)) * (1 - x^k)^5 / (1 - x^k + x^(2*k))^4.
a(n) = (-1)^n * A113261(n). Convolution of A123330 and A131943.
a(n) = -A132000(n) unless n=0.
Expansion of (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 03 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 15552^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A122373. - Michael Somos, Nov 03 2015

A326575 Expansion of Sum_{k>=1} k * x^k * (1 + x^(2*k)) / (1 + x^(2*k) + x^(4*k)).

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 8, 8, 9, 8, 10, 12, 14, 16, 12, 16, 16, 18, 20, 16, 24, 20, 22, 24, 21, 28, 27, 32, 28, 24, 32, 32, 30, 32, 32, 36, 38, 40, 42, 32, 40, 48, 44, 40, 36, 44, 46, 48, 57, 42, 48, 56, 52, 54, 40, 64, 60, 56, 58, 48, 62, 64, 72, 64, 56, 60
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 12 2019

Keywords

Examples

			G.f. = x + 2*x^2 + 3*x^3 + 4*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 8*x^8 + ... - _Michael Somos_, Oct 23 2019

Links

Crossrefs

Cf. A003586 (fixed points), A035178, A050469, A122373, A326401.
Cf. A175646, A340578.

Programs

  • Mathematica
    nmax = 66; CoefficientList[Series[Sum[k x^k (1 + x^(2 k))/(1 + x^(2 k) + x^(4 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, MemberQ[{1}, Mod[n/#, 6]] &] - DivisorSum[n, # &, MemberQ[{5}, Mod[n/#, 6]] &], {n, 1, 66}]
f[p_, e_] := Which[p < 5, p^e, Mod[p, 6] == 5, (p^(e + 1) - (-1)^(e + 1))/(p + 1), Mod[p, 6] == 1, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)
  • PARI
    a(n) = { sumdiv(n, d, d*((n/d%6==1)-(n/d%6==5))) } \\ Andrew Howroyd, Sep 12 2019
  • PARI
    {a(n) = if( n<0, 0, sumdiv( n, d, n/d * kronecker( -12, d)))}; /* Michael Somos, Oct 23 2019 */

Formula

a(n) = Sum_{d|n, n/d==1 (mod 6)} d - Sum_{d|n, n/d==5 (mod 6)} d.
G.f.: Sum_{k>=0} x^(6*k+1) / (1 - x^(6*k+1))^2 - x^(6*k+5) / (1 - x^(6*k+5))^2. - Michael Somos, Oct 23 2019
Multiplicative with a(p^e) = p^e if p < 5, (p^(e+1)-(-1)^(e+1))/(p+1) if p == 5 (mod 6), and (p^(e+1)-1)/(p-1) if p == 1 (mod 6). - Amiram Eldar, Dec 02 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{primes p == 5 (mod 6)} 1/(1+1/p^2) * Product_{primes p == 1 (mod 3)} 1/(1 - 1/p^2) = A340578 * A175646 / 2 = 0.48831400806... . - Amiram Eldar, Nov 06 2022
Showing 1-2 of 2 results.

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