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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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A106401 Expansion of (eta(q) * eta(q^9))^3 / eta(q^3)^2 in powers of q.

Original entry on oeis.org

1, -3, 0, 7, -6, 0, 8, -15, 0, 18, -12, 0, 14, -24, 0, 31, -18, 0, 20, -42, 0, 36, -24, 0, 31, -42, 0, 56, -30, 0, 32, -63, 0, 54, -48, 0, 38, -60, 0, 90, -42, 0, 44, -84, 0, 72, -48, 0, 57, -93, 0, 98, -54, 0, 72, -120, 0, 90, -60, 0, 62, -96, 0, 127, -84
Offset: 1

Views

Author

Michael Somos, May 02 2005

Keywords

Comments

Number 21 of the 74 eta-quotients listed in Table I of Martin (1996).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = q - 3*q^2 + 7*q^4 - 6*q^5 + 8*q^7 - 15*q^8 + 18*q^10 - 12*q^11 +...

Links

Crossrefs

Cf. A033686, A144614.
Cf. A004016, A005882, A005928.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(9), 2), 66); A[2] - 3*A[3]; /* Michael Somos, May 18 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^9])^3 / QPochhammer[ q^3]^2, {q, 0, n}]; (* Michael Somos, May 18 2015 *)
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^9 + A))^3 / eta(x^3 + A)^2, 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, 0, (-1)^((p%3>1) * e) * (p^(e+1) - 1) / (p - 1))))};

Formula

G.f.: x * Product_{k>0} (1 - x^k)^3 * (1 - x^(9*k))^3 / (1 - x^(3*k))^2.
Expansion of b(q) * c(q^3) / 3 in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Oct 17 2006
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = -v^3 + 6 * u*v*w + 4 * u*w^2 + u^2*w.
Euler transform of period 9 sequence [ -3, -3, -1, -3, -3, -1, -3, -3, -4, ...].
a(n) is multiplicative with a(3^e) = 0 if e>0, a(p^e) = (p^(e+1) - 1) / (p - 1) if e even or p == 1 (mod 3), a(p^e) = -(p^(e+1) - 1) / (p - 1) otherwise. - Michael, Somos Oct 19 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 9 (t/i)^2 f(t) where q = exp(2 Pi i t).
a(3*n) = 0. a(3*n + 1) = A144614(n). a(3*n + 2) = -3 * A033686(n).
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = 4*Pi^2/81 = 0.487387... . - Amiram Eldar, Jan 23 2024

A232356 Expansion of 2/9 * c(q) * c(q^2) - q * (psi(q) * psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function and c(q) is a cubic AGM theta function.

Original entry on oeis.org

1, 0, 5, -2, 6, 4, 8, -6, 17, 0, 12, 2, 14, 0, 30, -14, 18, 16, 20, -12, 40, 0, 24, -2, 31, 0, 53, -16, 30, 24, 32, -30, 60, 0, 48, 14, 38, 0, 70, -36, 42, 32, 44, -24, 102, 0, 48, -10, 57, 0, 90, -28, 54, 52, 72, -48, 100, 0, 60, 12, 62, 0, 136, -62, 84, 48
Offset: 1

Views

Author

Michael Somos, Nov 22 2013

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 + 5*q^3 - 2*q^4 + 6*q^5 + 4*q^6 + 8*q^7 - 6*q^8 + 17*q^9 + ...

Links

Crossrefs

Cf. A008438, A033686, A098098, A111932, A121443, A123532, A144614, A229615, A232343.

Programs

  • Magma
    Basis( ModularForms( Gamma0(6), 2), 70) [2];
  • Mathematica
    a[ n_] := If[ n < 1, 0, Sum[ d ( 2 Mod[ d, 2] Boole[Mod[ n/d, 3] > 0] - Mod[ n/d, 2] Boole[ Mod[d, 3] > 0]), {d, Divisors @n}]];
a[ n_] := SeriesCoefficient[ 2 q (QPochhammer[ q^3] QPochhammer[ q^6])^3 / (QPochhammer[ q] QPochhammer[ q^2]) - q (QPochhammer[ q^2] QPochhammer[ q^6])^4 / (QPochhammer[ q] QPochhammer[ q^3])^2, {q, 0, n}];
  • PARI
    {a(n) = local(A); if( n<1, 0, n--; A=x*O(x^n); polcoeff( 2 * (eta(x^3 + A) * eta(x^6 + A))^3 / (eta(x + A) * eta(x^2 + A)) - (eta(x^2 + A) * eta(x^6 + A))^4 / (eta(x + A) * eta(x^3 + A))^2, n))};
  • Sage
    ModularForms( Gamma0(6), 2, prec=70).1;

Formula

a(n) = 2 * A121443(n) - A111932(n). a(2*n) = -2 * A229615(n). a(12*n + 2) = a(12*n + 10) = 0.
a(n) = A123532(n) + 7 * A229615(n). a(3*n + 2) = 6 * A232343(n-1). a(6*n + 5) = 6 * A098098(n). a(12*n + 4) = -2 * A144614(n). a(12*n + 6) = 4 * A008438(n). a(12*n + 8) = -6 * A033686(n). - Michael Somos, May 23 2014

A295012 a(n) = sigma(12n - 1)/12, where sigma = sum of divisors (A000203).

Original entry on oeis.org

1, 2, 4, 4, 5, 6, 7, 10, 9, 12, 11, 14, 16, 14, 15, 16, 20, 22, 19, 20, 21, 22, 31, 28, 28, 26, 30, 34, 29, 30, 36, 32, 40, 38, 35, 36, 37, 56, 39, 40, 41, 42, 52, 48, 57, 50, 47, 62, 49, 50, 56, 60, 64, 54, 55, 62, 57, 70, 68, 60, 66, 62, 76, 70, 70, 76
Offset: 1

Views

Author

M. F. Hasler, Dec 08 2017

Keywords

Comments

Robert G. Wilson v observes in A280098 that {1, 3, 4, 6, 8, 12, 24} seem to be the only positive integers k such that sigma(kn-1)/k is an integer for all n > 0.

Links

Crossrefs

Cf. A280098 (analog for k = 24), A097723 (analog for k = 4), A033686 (analog for k = 3), A000203 (sigma, also the analog for k = 1).
The analog for k = 8 is A258835, up to the offset.
The analog for k = 6 is A098098 (up to the offset), a signed variant of this and the preceding one is A258831.
Cf. A086463.

Programs

  • GAP
    sequence := List([1..10^5], n-> Sigma(12 *n-1)/12); # Muniru A Asiru, Dec 28 2017
  • Maple
    with(numtheory):
seq(sigma(12*n-1)/12, n=1..10^3); # Muniru A Asiru, Dec 28 2017
  • Mathematica
    Array[DivisorSigma[1, 12 # - 1]/12 &, 66] (* Michael De Vlieger, Dec 08 2017 *)
  • PARI
    vector(90,n,sigma(12*n-1)/12)

Formula

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/18 = 0.548311... (A086463). - Amiram Eldar, Mar 28 2024
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