A111661 Expansion of eta(q)^4 * eta(q^2) * eta(q^6)^5 / eta(q^3)^4 in powers of q.

1, -4, 1, 16, -24, -4, 50, -64, 1, 96, -120, 16, 170, -200, -24, 256, -288, -4, 362, -384, 50, 480, -528, -64, 601, -680, 1, 800, -840, 96, 962, -1024, -120, 1152, -1200, 16, 1370, -1448, 170, 1536, -1680, -200, 1850, -1920, -24, 2112, -2208, 256, 2451, -2404, -288, 2720, -2808, -4

Offset

1,2

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).

References

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 226 Entry 4(i).

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Euler transform of period 6 sequence [-4, -5, 0, -5, -4, -6, ...].

Expansion of q * psi(q)^2 * phi(-q)^3 * psi(q^3)^2 / phi(-q^3) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Mar 01 2011

Expansion of (b(q^2)^3 - b(q)^3) / 9 in powers of q where b() is a cubic AGM theta function. - Michael Somos, Mar 01 2011

Expansion of (1/3) * b(q) * b(q^2) * c(q^2)^2 / c(q) in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Jul 09 2012

a(n) is multiplicative with a(2^e) = (-4)^e, a(3^e) = 1, a(p^e) = ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1 (mod 6), a(p^e) = (1 - (-p^2)^(e+1)) / (p^2 + 1) if p == 5 (mod 6). - Michael Somos, Mar 01 2011

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 243^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A214262.

G.f.: Sum_{k>0} Kronecker(k, 3) * k^2 * x^k / (1 - x^(2*k)) = x * Product_{k>0} (1 - x^k)^4 * (1 - x^(2*k)) * (1 + x^(3*k))^5 * (1 - x^(3*k)).

Example

G.f. = q - 4*q^2 + q^3 + 16*q^4 - 24*q^5 - 4*q^6 + 50*q^7 - 64*q^8 + q^9 + ...

Mathematica

a[ n_]:= If[ n < 1, 0, Sum[ Mod[ n/d, 2] d^2 KroneckerSymbol[d, 3], {d, Divisors[n]}]]; (* Michael Somos, Jul 09 2012 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[eta[q]^4* eta[q^2]*eta[q^6]^5/eta[q^3]^4, {q, 0, 30}], q] (* G. C. Greubel, Apr 18 2018 *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^2 * kronecker(d, 3)) )};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^2 + A) * eta(x^6 + A)^5 / eta(x^3 + A)^4, n))};

Crossrefs

Sequence in context: A059991 A002568 A334063 * A072651 A209411 A369912

Adjacent sequences: A111658 A111659 A111660 * A111662 A111663 A111664

Keyword

sign,mult

Author

Michael Somos, Aug 08 2005

Status

approved