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A256276 Expansion of q * phi(q) * chi(q^3) * psi(-q^9) in powers of q where phi(), psi(), chi() are Ramanujan theta functions. 4
1, 2, 0, 1, 4, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 1, 4, 0, 0, 4, 0, 0, 0, 0, 3, 4, 0, 0, 4, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 0, 1, 6, 0, 2, 4, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, 8, 0, 0, 4, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 4, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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 = 1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A256269.

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

EXAMPLE

G.f. = q + 2*q^2 + q^4 + 4*q^5 + 2*q^8 + 2*q^10 + 2*q^13 + q^16 + 4*q^17 + ...

MATHEMATICA

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

PROG

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^6 + A)^2 * eta(x^9 + A) * eta(x^36 + A) / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^3 + A) * eta(x^12 + A) * eta(x^18 + A)), n))};

(Magma) A := Basis( ModularForms( Gamma1(36), 1), 89); A[2] + 2*A[3]

+ A[5] + 4*A[6] + 2*A[9] + 2*A[11] + 2*A[14] + A[17] + 4*A[18];

CROSSREFS

Cf. A122856, A122865, A256269.

Sequence in context: A334173 A174996 A286815 * A257920 A258210 A258228

Adjacent sequences: A256273 A256274 A256275 * A256277 A256278 A256279

KEYWORD

nonn

AUTHOR

Michael Somos, Jun 02 2015

STATUS

approved

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Last modified November 30 06:18 EST 2022. Contains 358431 sequences. (Running on oeis4.)