login
A261203
Expansion of f(-x^6)^2 / (phi(-x) * phi(-x^9)) in powers of x where phi(), f() are Ramanujan theta functions.
2
1, 2, 4, 8, 14, 24, 38, 60, 92, 140, 208, 304, 439, 626, 884, 1232, 1704, 2336, 3182, 4300, 5772, 7700, 10212, 13472, 17673, 23076, 29988, 38808, 50008, 64184, 82070, 104560, 132760, 167996, 211920, 266512, 334202, 417902, 521152, 648224, 804254, 995432
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-1/2) * eta(q^2) * eta(q^6)^2 * eta(q^18) / (eta(q)^2 * eta(q^9)^2) in powers of q.
Euler transform of period 18 sequence [ 2, 1, 2, 1, 2, -1, 2, 1, 4, 1, 2, -1, 2, 1, 2, 1, 2, 0, ...].
a(n) = A261154(2*n + 1).
Convolution inverse of A261202.
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(11/4)*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017
EXAMPLE
G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 14*x^4 + 24*x^5 + 38*x^6 + 60*x^7 + ...
G.f. = q + 2*q^3 + 4*q^5 + 8*q^7 + 14*q^9 + 24*q^11 + 38*q^13 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^6]^2 / (EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^9]), {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)^2 * eta(x^18 + A) / (eta(x + A)^2 * eta(x^9 + A)^2), n))};
CROSSREFS
Sequence in context: A100250 A053800 A262968 * A281968 A091774 A344741
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 11 2015
STATUS
approved