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A328796 Expansion of chi(x) / chi(-x^6) in powers of x where chi() is a Ramanujan theta function. 3
1, 1, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5, 7, 8, 8, 11, 12, 12, 16, 17, 18, 23, 25, 26, 32, 35, 37, 45, 49, 52, 62, 67, 72, 85, 92, 98, 114, 124, 133, 153, 166, 178, 203, 220, 236, 268, 290, 311, 350, 379, 407, 456, 493, 529, 589, 636, 683, 758, 818, 877 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

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

Convolution square is A328790.

G.f. is a period 1 Fourier series which satisfies f(-1 / (1728 t)) = 2^(-1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A328880.

LINKS

Table of n, a(n) for n=0..62.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q^(-5/24) * (eta(q^2)^2 * eta(q^12)) / (eta(q) * eta(q^4) * eta(q^6)) in power of q.

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

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

A261736(n) = (-1)^n * a(n).

a(n) ~ exp(sqrt(2*n)*Pi/3) / (2^(7/4)*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Oct 31 2019

EXAMPLE

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

G.f. = q^5 + q^29 + q^77 + q^101 + q^125 + 2*q^149 + 2*q^173 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^6, x^6], {x, 0, n}];

PROG

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

CROSSREFS

Cf. A261736, A328790, A328800.

Sequence in context: A179269 A108711 A261736 * A247049 A029059 A035449

Adjacent sequences:  A328793 A328794 A328795 * A328797 A328798 A328799

KEYWORD

nonn

AUTHOR

Michael Somos, Oct 27 2019

STATUS

approved

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Last modified February 25 05:11 EST 2020. Contains 332217 sequences. (Running on oeis4.)