A115110 Expansion of q^(-1/24) * eta(q)^3 / eta(q^2) in powers of q.

1, -3, 1, 2, 2, -1, -4, 1, -2, 0, 2, 4, -1, 2, -2, -1, 0, -2, -2, -2, 0, 4, 1, 0, 2, -2, 5, 0, -2, 0, 0, -4, -2, 0, 0, -3, 4, 0, 0, -2, 1, 4, 2, 2, 0, 0, 0, -2, -2, 0, 2, -3, -2, 0, -2, 2, -4, 1, 0, 0, 0, 4, 2, 0, 4, 0, -4, 2, 0, 2, -1, 0, 0, 2, -2, -2, -6, -1, 2, 0, 0, -4, 0, 2, 2, 0, 0, 2, -2, 2, 2, 0, 1, 0, 0, 2, 4, 0, 0, -2, 1, -6, 0, -2, 0

Offset

0,2

Comments

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

References

B. Gordon and D. Sinor, Multiplicative properties of eta-products, Number theory, Madras 1987, pp. 173-200, Lecture Notes in Math., 1395, Springer, Berlin, 1989; see page 182. MR1019331 (90k:11050)

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134; see page 124 (5.15).

Peter Bala, Identities for A115110

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of f(x) * f(-x) in powers of x^2 where f() is a Ramanujan theta function.

Expansion of f(-x) * phi(-x) in powers of x where phi(), f() are Ramanujan theta functions.

Given A = A0 + A1 + A2 + A3 + A4 + A5 + A6 is the 7-section, then 0 = A0*A4 + A1*A3 + A5*A6 + 4*A2^2, A2 = x^2 * A(x^49).

Euler transform of period 2 sequence [ -3, -2,...].

G.f.: Product_{k>0} (1 - x^k)^2 / (1 + x^k).

G.f.: Sum_{k>=0} ( x^((3*k^2 + k)/2) * (1 - x^(2*k + 1)) * Sum_{|j|<=k} (-x)^(-j^2) ).

a(49*n + 2) = a(n). a(7*n + 2) = 0 unless n = 7*k.

a(n) = (-1)^n * A107033(n).

G.f.: exp( Sum_{n>=1} -sigma(2*n)*x^n/n ). - Seiichi Manyama, Mar 02 2017

a(n) = -(1/n)*Sum_{k=1..n} sigma(2*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

From Peter Bala, Jan 01 2021: (Start)

For prime p of the form 4*k + 3, a(n*p^2 + (p^2 - 1)/24) = e*a(n), where e = 1 if p == 7 or 23 (mod 24) and e = -1 if p == 11 or 19 (mod 24).

If n > 0 and p are coprime then a(n*p + (p^2 - 1)/24) = 0. Cf. A002107.

(End)

Example

G.f. = 1 - 3*x + x^2 + 2*x^3 + 2*x^4 - x^5 - 4*x^6 + x^7 - 2*x^8 + 2*x^10 + ...
G.f. = q - 3*q^25 + q^49 + 2*q^73 + 2*q^97 - q^121 - 4*q^145 + q^169 - 2*q^193 + ...

Maple

prod := n -> mul( (1 - x^k)^2*(1 - x^(2*k-1)), k = 1..n):
a := n -> coeff(prod(100), x, n):
seq(a(n), n = 0..100); # Peter Bala, Jan 01 2021

Mathematica

a[ n_] :=  SeriesCoefficient[ QPochhammer[ x]^3 / QPochhammer[ x^2], {x, 0, n}]; (* Michael Somos, Jul 12 2012 *)
a[ n_] :=  SeriesCoefficient[ QPochhammer[ x] QPochhammer[ -x], {x, 0, 2 n}]; (* Michael Somos, Jul 12 2012 *)
a[ n_] :=  SeriesCoefficient[ QPochhammer[ x] EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Jul 12 2012 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 / eta(x^2 + A), n))};
(Magma) m:=120; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^j)^2 / (1 + x^j): j in [1..m+2]]) )); // G. C. Greubel, Nov 18 2018
(Sage)
R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(120)
s = prod((1 - x^j)^2 / (1 + x^j) for j in (1..120))
s.coefficients() # G. C. Greubel, Nov 18 2018

Crossrefs

Cf. A107033, A002107.

Cf. Product_{n>=1} (1 - q^n)^(k+1)/(1 - q^(k*n)): A010815 (k=1), this sequence (k=2), A185654 (k=3), A282937 (k=5), A282942 (k=7).

Sequence in context: A111951 A222593 A107033 * A066635 A016568 A327314

Adjacent sequences: A115107 A115108 A115109 * A115111 A115112 A115113

Keyword

sign,easy

Author

Michael Somos, Mar 07 2006

Status

approved