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A261118 Expansion of psi(x)^2 * psi(-x^3)^2 / (phi(-x^4) * psi(-x^6)) in power of x where phi(), psi() are Ramanujan theta functions. 4
1, 2, 1, 0, 0, 2, 3, 2, 2, 0, 0, 2, 3, 2, 0, 0, 0, 0, 2, 4, 1, 0, 0, 2, 2, 2, 4, 0, 0, 0, 3, 4, 0, 0, 0, 0, 4, 2, 0, 0, 0, 4, 1, 2, 2, 0, 0, 2, 2, 2, 0, 0, 0, 0, 4, 0, 3, 0, 0, 2, 2, 6, 2, 0, 0, 2, 4, 2, 0, 0, 0, 0, 1, 2, 2, 0, 0, 2, 2, 2, 2, 0, 0, 0, 2, 4, 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

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 f(-x^8) * f(x, x^5)^2 / psi(-x^6) in powers of x where psi(), f() are Ramanujan theta functions.

Expansion of q^(-1/4) * eta(q^2)^4 * eta(q^3)^2 * eta(q^8) * eta(q^12)^3 / (eta(q)^2 * eta(q^4)^2 * eta(q^6)^3 * eta(q^24)) in powers of q.

Euler transform of period 24 sequence [ 2, -2, 0, 0, 2, -1, 2, -1, 0, -2, 2, -2, 2, -2, 0, -1, 2, -1, 2, 0, 0, -2, 2, -2, ...].

a(n) = (-1)^n * A259668(n) = A129402(2*n) = A190615(2*n) = A192013(4*n) = A000377(4*n + 1) = A129402(6*n + 1).

a(2*n) = A260308(n). a(2*n + 1) = 2 * A259895(n).

EXAMPLE

G.f. = 1 + 2*x + x^2 + 2*x^5 + 3*x^6 + 2*x^7 + 2*x^8 + 2*x^11 + 3*x^12 + ...

G.f. = q + 2*q^5 + q^9 + 2*q^21 + 3*q^25 + 2*q^29 + 2*q^33 + 2*q^45 + ...

MATHEMATICA

a[n_]:= SeriesCoefficient[(-1)^(-1/8)*q^(-1/4)*(EllipticTheta[2, 0, Sqrt[q]]*EllipticTheta[2, 0, I*Sqrt[q^3]])^2/(8*EllipticTheta[3, 0, -q^4]*EllipticTheta[2, 0, I*q^3]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jan 04 2018 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^3 + A)^2 * eta(x^8 + A) * eta(x^12 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^3 * eta(x^24 + A)), n))};

CROSSREFS

Cf. A129402, A190615, A192013, A259668, A259895, A260308.

Sequence in context: A060701 A275345 A259668 * A063181 A247310 A059220

Adjacent sequences:  A261115 A261116 A261117 * A261119 A261120 A261121

KEYWORD

nonn

AUTHOR

Michael Somos, Aug 08 2015

STATUS

approved

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Last modified November 27 12:51 EST 2021. Contains 349394 sequences. (Running on oeis4.)