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A145703
Expansion of chi(x) / chi(-x^10) in powers of x where chi() is a Ramanujan theta function.
4
1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 6, 6, 7, 8, 10, 11, 11, 13, 15, 17, 18, 20, 23, 25, 29, 32, 34, 39, 42, 47, 52, 56, 62, 68, 77, 83, 89, 99, 108, 119, 129, 139, 154, 167, 183, 199, 214, 234, 253, 276, 299, 322, 350, 378, 413, 445, 476, 518, 559
OFFSET
0,9
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^(-3/8) * eta(q^2)^2 * eta(q^20) / (eta(q) * eta(q^4) * eta(q^10) ) in powers of q.
Euler transform of period 20 sequence [ 1, -1, 1, 0, 1, -1, 1, 0, 1, 0, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (640 t)) = 2^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A145702.
G.f.: Product_{k>0} (1 + x^(2*k - 1)) / (1 - x^(20*k - 10)).
a(n) = (-1)^n * A145707(n) = A139632(2*n + 1).
a(n) ~ exp(Pi*sqrt(n/5)) / (4*5^(1/4)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015
EXAMPLE
G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + 3*x^10 + ...
G.f. = q^3 + q^11 + q^27 + q^35 + q^43 + q^51 + q^59 + 2*q^67 + 2*q^75 + ...
MATHEMATICA
nmax = 100; CoefficientList[Series[Product[(1 + x^(2*k-1)) / (1 - x^(20*k-10)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^10, x^10], {x, 0, n}]; (* Michael Somos, Sep 06 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^20 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^10 + A)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 17 2008
STATUS
approved