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A187153 Expansion of q * (psi(q) / psi(q^2)) / (psi(q^3) / psi(q^6))^3 in powers of q where psi() is a Ramanujan theta function. 5
1, 1, -1, -3, -2, 3, 8, 5, -7, -18, -12, 15, 38, 24, -30, -75, -46, 57, 140, 86, -104, -252, -152, 183, 439, 262, -313, -744, -442, 522, 1232, 725, -852, -1998, -1168, 1365, 3182, 1852, -2150, -4986, -2886, 3336, 7700, 4436, -5106, -11736, -6736, 7719 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

W. Zudilin, Deciding whether a given power series is modular or not

FORMULA

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

Euler transform of period 12 sequence [ 1, -2, -2, 0, 1, 4, 1, 0, -2, -2, 1, 0, ...].

Expansion of c(q) * c(q^4)^2 / c(q^2)^3 in powers of q where c() is a cubic AGM theta function.

If p = 2 * A(q), then B(q) = p * ((2 + p) / (1 + 2*p))^3 and B(q^3) = p^3 * ((2 + p) / (1 + 2*p)) where B() is the g.f. for A115977. - Michael Somos, Feb 27 2012

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

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^2 * (1 + 4*v) * (3*v + u*(1 + 4*v)) - v * (1 + v) * (3*u + 1 + v).

Convolution inverse of A187143.

EXAMPLE

G.f. = q + q^2 - q^3 - 3*q^4 - 2*q^5 + 3*q^6 + 8*q^7 + 5*q^8 - 7*q^9 - 18*q^10 + ...

MATHEMATICA

QP = QPochhammer; s = (QP[q^2]*QP[q^3]*QP[q^12]^2)^3/(QP[q]*QP[q^4]^2* QP[q^6]^9) + O[q]^50; CoefficientList[s, q] (* Jean-Fran├žois Alcover, Nov 14 2015, adapted from PARI *)

QP = QPochhammer; Rest[Table[SeriesCoefficient[q*(QP[-q, q^2]*QP[-q^6, q^6]^3)/(QP[-q^2, q^2]*QP[-q^3, q^6]^3), {q, 0, n}], {n, 0, 50}]] (* G. C. Greubel, Dec 04 2017 *)

PROG

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

CROSSREFS

Cf. A187143.

Sequence in context: A070982 A275520 * A213265 A123649 A080848 A016602

Adjacent sequences:  A187150 A187151 A187152 * A187154 A187155 A187156

KEYWORD

sign,changed

AUTHOR

Michael Somos, Mar 06 2011

STATUS

approved

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Last modified December 15 03:08 EST 2017. Contains 296020 sequences.