A208955 Expansion of phi(x) * phi(x^9) / chi(x^3)^2 in powers of x where phi(), chi() are Ramanujan theta functions.

1, 2, 0, -2, -2, 0, 3, 2, 0, -2, -2, 0, 1, 2, 0, -2, -4, 0, 2, 0, 0, -4, -2, 0, 2, 2, 0, -2, -2, 0, 1, 4, 0, 0, -2, 0, 4, 2, 0, -2, 0, 0, 3, 2, 0, -2, -4, 0, 2, 2, 0, -4, 0, 0, 0, 4, 0, -2, -2, 0, 2, 2, 0, 0, -2, 0, 4, 2, 0, -2, -2, 0, 3, 2, 0, 0, -4, 0, 2, 2

Offset

0,2

Comments

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

Number 74 of the 74 eta-quotients listed in Table I of Martin (1996).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/4) * eta(q^2)^5 * eta(q^3)^2 * eta(q^12)^2 * eta(q^18)^5 / (eta(q) * eta(q^4) * eta(q^6)^2 * eta(q^9) * eta(q^36))^2 in powers of q.

Euler transform of period 36 sequence [ 2, -3, 0, -1, 2, -1, 2, -1, 2, -3, 2, -1, 2, -3, 0, -1, 2, -4, 2, -1, 0, -3, 2, -1, 2, -3, 2, -1, 2, -1, 2, -1, 0, -3, 2, -2, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 24 (t/i) f(t) where q = exp(2 Pi i t).

a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1, 5 (mod 24), b(p^e) = (e+1) * (-1)^e if p == 13, 17 (mod 24).

a(n) = (-1)^n * A121363(n). a(3*n + 2) = 0.

Example

G.f. = 1 + 2*x - 2*x^3 - 2*x^4 + 3*x^6 + 2*x^7 - 2*x^9 - 2*x^10 + x^12 + ...
G.f. = q + 2*q^5 - 2*q^13 - 2*q^17 + 3*q^25 + 2*q^29 - 2*q^37 - 2*q^41 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 3, 0, x^9] / QPochhammer[ -x^3, x^6]^2 , {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)
a[ n_] := If[ n < 0, 0, (-1)^n With[ {m = 4 n + 1}, DivisorSum[ m, KroneckerSymbol[ 12, #] KroneckerSymbol[ 12, m/#] &]]]; (* Michael Somos, Jun 09 2015 *)

Prog

(PARI) {a(n) = my(m); if( n<0, 0, m = 4*n + 1; (-1)^n * dirmul( vector(m, k, kronecker( 12, k)), vector(m, k, kronecker( 12, k)))[m])};
(PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 4*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 0, p%4==3, (1 + (-1)^e)/2, (e+1) * if( p%24>12, (-1)^e, 1)))) };
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^2 * eta(x^12 + A)^2 * eta(x^18 + A)^5 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^9 + A) * eta(x^36 + A))^2, n))};

Crossrefs

Cf. A121363.

Sequence in context: A329860 A334209 A143613 * A121363 A346274 A214258

Adjacent sequences: A208952 A208953 A208954 * A208956 A208957 A208958

Keyword

sign

Author

Michael Somos, Mar 03 2012

Status

approved