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A145722
Expansion of f(q) * f(q^5) / phi(-q^2)^2 in powers of q where f(), phi() are Ramanujan theta functions.
5
1, 1, 3, 4, 8, 12, 21, 30, 48, 68, 102, 143, 207, 284, 400, 542, 744, 996, 1344, 1776, 2361, 3088, 4050, 5248, 6808, 8742, 11232, 14310, 18224, 23052, 29133, 36601, 45936, 57360, 71528, 88812, 110110, 135990, 167704, 206108, 252912, 309408
OFFSET
0,3
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
MathOverflow, Up to 2000..., 2016.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-1/4) * eta(q^4) * eta(q^10)^3 / (eta(q) * eta(q^2) * eta(q^5) * eta(q^20)) in powers of q.
Euler transform of period 20 sequence [ 1, 2, 1, 1, 2, 2, 1, 1, 1, 0, 1, 1, 1, 2, 2, 1, 1, 2, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 20^(-1/2) g(t), where q = exp(2 Pi i t) and g() is the g.f. for A145723.
G.f.: Product_{k>0} (1 + x^(2*k)) * (1 - x^(10*k)) * (1 + x^(5*k)) / ((1 - x^k) * (1 + x^(10*k))).
a(n) = A036026(2*n).
a(n) ~ exp(2*Pi*sqrt(n/5)) / (4 * 5^(3/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015
EXAMPLE
G.f. = 1 + x + 3*x^2 + 4*x^3 + 8*x^4 + 12*x^5 + 21*x^6 + 30*x^7 + 48*x^8 + ...
G.f. = q + q^5 + 3*q^9 + 4*q^13 + 8*q^17 + 12*q^21 + 21*q^25 + 30*q^29 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ -x^5] / EllipticTheta[ 4, 0, x^2]^2, {x, 0, n}]; (* Michael Somos, Aug 26 2015 *)
nmax=60; CoefficientList[Series[Product[(1+x^(2*k)) * (1-x^(10*k)) * (1+x^(5*k)) / ((1-x^k) * (1 + x^(10*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^10 + A)^3 / (eta(x + A) * eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A)), n))};
CROSSREFS
Cf. A036026.
Sequence in context: A281612 A349050 A025034 * A147622 A173534 A074331
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 23 2008
STATUS
approved