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A134004 Expansion of (chi(-x) * chi(-x^19))^(-2) in powers of x where chi() is a Ramanujan theta function. 1
1, 2, 3, 6, 9, 14, 22, 32, 46, 66, 93, 128, 176, 238, 319, 426, 562, 736, 960, 1244, 1602, 2054, 2620, 3324, 4203, 5292, 6634, 8290, 10322, 12808, 15845, 19542, 24028, 29468, 36042, 43966, 53506, 64960, 78685, 95106, 114709, 138066, 165855, 198856, 237979 (list; graph; refs; listen; history; text; internal format)
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^(-5/3) * (eta(q^2) * eta(q^38) / (eta(q) * eta(q^19)))^2 in powers of q.
Euler transform of period 38 sequence [ 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (342 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134005.
G.f.: (Product_{k>0} (1 + x^k) * (1 + x^(19*k)))^2.
Convolution inverse of A134005.
a(n) ~ exp(2*Pi*sqrt(10*n/57)) * 5^(1/4) / (2^(11/4) * 57^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
EXAMPLE
G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 9*x^4 + 14*x^5 + 22*x^6 + 32*x^7 + 46*x^8 + ...
G.f. = q^5 + 2*q^8 + 3*q^11 + 6*q^14 + 9*q^17 + 14*q^20 + 22*q^23 + 32*q^26 + ...
MATHEMATICA
nmax = 40; CoefficientList[Series[Product[((1 + x^k) * (1 + x^(19*k)))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x] QPochhammer[ -x^19, x^19])^2, {x, 0, n}]; (* Michael Somos, Oct 30 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^38 + A) / (eta(x + A) * eta(x^19 + A)))^2, n))};
CROSSREFS
Cf. A134005.
Sequence in context: A333697 A128518 A022567 * A123631 A228364 A018060
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 01 2007
STATUS
approved

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)