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A258327 Expansion of phi(x^3) / f(-x^2) in powers of x where phi(), f() are Ramanujan theta functions. 2
1, 0, 1, 2, 2, 2, 3, 4, 5, 6, 7, 10, 13, 14, 17, 22, 26, 30, 36, 44, 52, 60, 70, 84, 99, 112, 131, 156, 179, 204, 236, 274, 315, 358, 409, 472, 539, 608, 692, 792, 897, 1010, 1144, 1298, 1464, 1644, 1849, 2088, 2347, 2622, 2940, 3304, 3694, 4118, 4600, 5142 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

LINKS

Vaclav Kotesovec, 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, x^2) / psi(-x) in powers of x where psi(), f() are Ramanujan theta functions.

Expansion of q^(-1/12) * eta(q^6)^5 / (eta(q^2) * eta(q^3)^2 * eta(q^12)^2) in powers of q.

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

G.f.: Product_{k>0} (1 + x^k)^2 * (1 - x^k + x^(2*k))^3 * (1 + x^k + x^(2*k)) / (1 + x^(6*k))^2.

a(n) = (-1)^n * A256636(n).

a(n) ~ exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Jul 11 2016

EXAMPLE

G.f. = 1 + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 6*x^9 + ...

G.f. = 1/q + q^23 + 2*q^35 + 2*q^47 + 2*q^59 + 3*q^71 + 4*q^83 + 5*q^95 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3] / QPochhammer[ x^2], {x, 0, n}];

nmax = 50; CoefficientList[Series[Product[(1+x^(6*k-3)) / ((1-x^(6*k-2)) * (1-x^(6*k-3)) * (1-x^(6*k-4)) * (1+x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 11 2016 *)

PROG

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

CROSSREFS

Cf. A256636.

Sequence in context: A118301 A018121 A256636 * A102240 A026837 A005855

Adjacent sequences:  A258324 A258325 A258326 * A258328 A258329 A258330

KEYWORD

nonn

AUTHOR

Michael Somos, May 26 2015

STATUS

approved

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Last modified April 16 07:59 EDT 2021. Contains 343030 sequences. (Running on oeis4.)