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A226225 Expansion of phi(q) * phi(q^8) in powers of q where phi() is a Ramanujan theta function. 3
1, 2, 0, 0, 2, 0, 0, 0, 2, 6, 0, 0, 4, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 2, 8, 0, 0, 6, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 6, 4, 0, 0, 4, 0, 0, 0, 0, 10, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2500 from G. C. Greubel)

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of (eta(q^2) * eta(q^16))^5 / (eta(q) * eta(q^4) * eta(q^8) * eta(q^32))^2 in powers of q.

Euler transform of period 32 sequence [2, -3, 2, -1, 2, -3, 2, 1, 2, -3, 2, -1, 2, -3, 2, -4, 2, -3, 2, -1, 2, -3, 2, 1, 2, -3, 2, -1, 2, -3, 2, -2, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(1/2) (t/i) f(t) where q = exp(2 Pi i t).

G.f.: (Sum_{k in Z} x^k^2) * (Sum_{k in Z} (x^8)^k^2).

a(4*n + 2) = a(4*n + 3) = a(8*n + 5) = 0.  a(4*n) = a(8*n) = A033715(n). a(4*n + 1) = A033715(4*n + 1). a(8*n + 1) = 2 * A112603(n). a(8*n + 4) = 2 * A113411(n).

(-1)^n * a(n) = A242609(n). - Michael Somos, Feb 20 2015

EXAMPLE

G.f. = 1 + 2*q + 2*q^4 + 2*q^8 + 6*q^9 + 4*q^12 + 2*q^16 + 4*q^17 + 4*q^24 + 2*q^25 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^8], {q, 0, n}];

PROG

(PARI) {a(n) = if( n<1, n==0, 2 * (n%4 < 2) * sumdiv( n, d, kronecker( -2, d)))};

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^16 + A))^5 / (eta(x + A) * eta(x^4 + A) * eta(x^8 + A) * eta(x^32 + A))^2, n))};

CROSSREFS

Cf. A033715, A112603, A113411, A242609.

Sequence in context: A108498 A178923 A242609 * A329265 A130209 A109127

Adjacent sequences:  A226222 A226223 A226224 * A226226 A226227 A226228

KEYWORD

nonn

AUTHOR

Michael Somos, May 31 2013

STATUS

approved

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Last modified March 30 19:49 EDT 2020. Contains 333127 sequences. (Running on oeis4.)