A182057 Expansion of psi(x) * f(x^4) / (psi(x^3) * f(x^6) * chi(-x^24)) in powers of x where psi(), chi(), f() are Ramanujan theta functions.

1, 1, 0, 0, 0, 1, 0, 0, -2, -3, 0, 0, 4, 1, 0, 0, -1, 5, 0, 0, -8, -10, 0, 0, 14, 4, 0, 0, -4, 17, 0, 0, -23, -31, 0, 0, 40, 9, 0, 0, -10, 46, 0, 0, -60, -79, 0, 0, 98, 21, 0, 0, -24, 112, 0, 0, -140, -183, 0, 0, 224, 46, 0, 0, -54, 249, 0, 0, -304, -396, 0

Offset

0,9

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 q^(-2/3) * eta(q^2)^2 * eta(q^3) * eta(q^8)^3 * eta(q^48) / (eta(q) * eta(q^4) * eta(q^6) * eta(q^12)^3 * eta(q^16)) in powers of q.

Euler transform of period 48 sequence [1, -1, 0, 0, 1, -1, 1, -3, 0, -1, 1, 3, 1, -1, 0, -2, 1, -1, 1, 0, 0, -1, 1, 0, 1, -1, 0, 0, 1, -1, 1, -2, 0, -1, 1, 3, 1, -1, 0, -3, 1, -1, 1, 0, 0, -1, 1, 0, ...].

a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A182032(12*n + 2). a(4*n + 1) = A182032(12*n + 5).

Expansion of psi(x) * f(x^4) / (f(x^3) * phi(x^6) * chi(x^12)) in powers of x where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Aug 10 2017

Example

G.f. = 1 + x + x^5 - 2*x^8 - 3*x^9 + 4*x^12 + x^13 - x^16 + 5*x^17 - 8*x^20 + ...
G.f. = q^2 + q^5 + q^17 - 2*q^26 - 3*q^29 + 4*q^38 + q^41 - q^50 + 5*q^53 + ...

Mathematica

eta[x_] := x^(1/24)*QPochhammer[x]; A182057[n_] := SeriesCoefficient[ q^(-2/3)*eta[q^2]^2*eta[q^3]*eta[q^8]^3*eta[q^48]/(eta[q]*eta[q^4]* eta[q^6]*eta[q^12]^3 *eta[q^16]), {q, 0, n}]; Table[A182057[n], {n, 0, 50}] (* G. C. Greubel, Aug 10 2017 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^8 + A)^3 * eta(x^48 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^12 + A)^3 * eta(x^16 + A)), n))}

Crossrefs

Cf. A182032.

Sequence in context: A011023 A284610 A234017 * A260928 A097027 A072741

Adjacent sequences: A182054 A182055 A182056 * A182058 A182059 A182060

Keyword

sign

Author

Michael Somos, Apr 08 2012

Status

approved