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A256269 Expansion of psi(-q) * chi(q^3) * phi(q^9) in powers of q where phi(), psi(), chi() are Ramanujan theta functions. 6
1, -1, 0, 0, -1, 0, 0, 0, 0, 4, -2, 0, 0, -2, 0, 0, -1, 0, 4, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 4, -2, 0, 0, -2, 0, 0, 0, 0, 8, 0, 0, 0, -1, 0, 0, -2, 0, 0, 0, 0, 0, -2, 0, 0, -2, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 4, -2, 0, 0, 0, 0, 0, 0, 0, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,10

COMMENTS

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

LINKS

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

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

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

a(3*n + 1) = - A122865(n). a(6*n + 4) = - A122856(n). a(3*n + 2) = a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0.

EXAMPLE

G.f. = 1 - q - q^4 + 4*q^9 - 2*q^10 - 2*q^13 - q^16 + 4*q^18 - 3*q^25 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(1/2)] / (2^(1/2) q^(1/8)) QPochhammer[ -q^3, q^6] EllipticTheta[ 3, 0, q^9], {q, 0, n}];

PROG

(PARI) {a(n) = if( n<1, n==0, (-1)^(n%3) * (n%3<2) * sumdiv(n, d, [ 0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^18 + A)^5 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A) * eta(x^9 + A)^2 * eta(x^36 + A)^2), n))};

(Magma) A := Basis( ModularForms( Gamma1(36), 1), 82); A[1] - A[2] - A[5] + 4*A[10] - 2*A[11] - 2*A[14] - A[17] + 4*A[19];

CROSSREFS

Cf. A122856, A122865.

Sequence in context: A159257 A258997 A232833 * A256279 A277767 A107088

Adjacent sequences: A256266 A256267 A256268 * A256270 A256271 A256272

KEYWORD

sign

AUTHOR

Michael Somos, Jun 01 2015

STATUS

approved

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Last modified November 30 05:38 EST 2022. Contains 358431 sequences. (Running on oeis4.)