

A298420


Expansion of f(x, x) * f(x, x^2) * f(x^2, x^4) in powers of x where f(, ) is Ramanujan's general theta function.


2



1, 3, 4, 5, 6, 8, 9, 6, 8, 8, 9, 12, 8, 12, 8, 13, 18, 8, 16, 12, 16, 13, 6, 16, 8, 18, 20, 16, 15, 12, 24, 18, 16, 16, 12, 20, 17, 18, 16, 16, 30, 24, 16, 12, 16, 21, 24, 16, 16, 18, 20, 32, 18, 28, 24, 27, 20, 16, 24, 12, 32, 30, 24, 16, 18, 32, 25, 18, 32
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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
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions


FORMULA

Expansion of phi(x^2) * phi(x^3) * phi(x^6) / chi(x)^3 in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of q^(1/4) * eta(q^2)^5 * eta(q^3)^2 * eta(q^6) / (eta(q)^3 * eta(q^4) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [3, 2, 1, 1, 3, 5, 3, 1, 1, 2, 3, 3, ...].
G.f. is a period 1 Fourier series which satisfies f(1 / (96 t)) = 48^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A298421.


EXAMPLE

G.f. = 1 + 3*x + 4*x^2 + 5*x^3 + 6*x^4 + 8*x^5 + 9*x^6 + 6*x^7 + 8*x^8 + ...
G.f. = q + 3*q^9 + 4*q^17 + 5*q^25 + 6*q^33 + 8*q^41 + 9*q^49 + 6*q^57 + ...


MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ q] QPochhammer[ q, q]^2 QPochhammer[ q^3]^2 QPochhammer[ q^6, q^12], {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2] EllipticTheta[ 4, 0, x^3] EllipticTheta[ 4, 0, x^6] QPochhammer[ x, x]^3, {x, 0, n}];


PROG

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


CROSSREFS

Cf. A298421.
Sequence in context: A096127 A327953 A112768 * A197354 A089399 A003619
Adjacent sequences: A298417 A298418 A298419 * A298421 A298422 A298423


KEYWORD

nonn


AUTHOR

Michael Somos, Jan 18 2018


STATUS

approved



