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A138522 Expansion of f(q, q^3)^2 / (f(q, q^4) * f(q^2, q^3)) in powers of q where f(, ) is the Ramanujan general theta function. 4
1, 1, -1, 0, -1, 1, 4, -4, -1, -3, 3, 12, -12, -2, -8, 8, 31, -30, -5, -20, 19, 72, -68, -12, -44, 41, 154, -144, -24, -90, 84, 312, -289, -48, -178, 164, 603, -554, -92, -336, 307, 1122, -1024, -168, -612, 557, 2024, -1836, -300, -1087, 983, 3552, -3206, -522 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

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

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of (eta(q^2) / eta(q^5))^3 * eta(q^10) / eta(q) in powers of q.

Euler transform of period 10 sequence [ 1, -2, 1, -2, 4, -2, 1, -2, 1, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v)^2 - u * (u + 4) * (1 - v).

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u-v)^4 - u * (1 - u) * (4 + u) * v * (1 - v) * (4 + v).

G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = (5/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138520.

G.f.: Product_{k>0} P(2, x^k)^4 * P(10, x^k) / P(5, x^k)^2 where P(n, x) is the n-th cyclotomic polynomial.

A095813(n) = a(n) unless n=0. Convolution inverse of A138520.

EXAMPLE

G.f. = 1 + q - q^2 - q^4 + q^5 + 4*q^6 - 4*q^7 - q^8 - 3*q^9 + 3*q^10 + 12*q^11 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] / QPochhammer[ q^5])^3 QPochhammer[ q^10] / QPochhammer[ q], {q, 0, n}]; (* Michael Somos, Sep 14 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^10 + A) / eta(x + A) * ( eta(x^2 + A) / eta(x^5 + A) )^3, n))};

CROSSREFS

Cf. A095813, A138520.

Sequence in context: A325489 A278516 A292434 * A095813 A010656 A321591

Adjacent sequences:  A138519 A138520 A138521 * A138523 A138524 A138525

KEYWORD

sign

AUTHOR

Michael Somos, Mar 23 2008

STATUS

approved

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Last modified March 30 12:10 EDT 2020. Contains 333125 sequences. (Running on oeis4.)