A102315 -id:A102315

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A328798

Expansion of 1 / (chi(-x) * chi(-x^3)) in powers of x where chi() is a Ramanujan theta function.

+10
4

1, 1, 1, 3, 3, 4, 7, 8, 10, 16, 19, 23, 33, 39, 48, 65, 77, 93, 122, 144, 173, 220, 259, 309, 384, 451, 534, 653, 764, 899, 1085, 1264, 1479, 1765, 2048, 2385, 2820, 3260, 3778, 4432, 5105, 5891, 6864, 7879, 9056, 10491, 12002, 13744, 15839, 18064, 20616, 23648

(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).

Convolution inverse is A112175, 2nd power is A102315, 3rd power is A229180, 6th power is A123653.

f(-1 / (216 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A112175.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

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

Euler transform of period 6 sequence [1, 0, 2, 0, 1, 0, ...].

G.f.: Product_{k>=1} (1 + x^k)^(-1) * (1 + x^(3*k))^(-1).

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

EXAMPLE

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

G.f. = q + q^7 + q^13 + 3*q^19 + 3*q^25 + 4*q^31 + 7*q^37 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ -x^3, 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) * eta(x^6 + A) / (eta(x + A) * eta(x^3 + A)), n))};

CROSSREFS

Cf. A102315, A112175, A123653, A229180.

KEYWORD

nonn

AUTHOR

Michael Somos, Oct 28 2019

STATUS

approved

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