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A134415
Expansion of phi(x) / f(-x)^6 in powers of x where phi(), f() are Ramanujan theta functions..
3
1, 8, 39, 152, 513, 1560, 4382, 11552, 28899, 69168, 159372, 355224, 768885, 1621296, 3339201, 6732232, 13311450, 25854744, 49398043, 92953016, 172451760, 315744072, 570997539, 1020691248, 1804730732, 3158323272, 5473566645, 9398873032, 15998363307, 27005721648
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^(1/4) * eta(q^2)^5 / (eta(q)^8 * eta(q^4)^2) in powers of q.
Euler transform of period 4 sequence [ 8, 3, 8, 5, ...].
G.f.: Product_{k>0} (1 + x^k)^3 / ((1 - x^k)^5 * (1 + x^(2*k))^2).
a(n) = A134414(4*n - 1).
a(n) ~ exp(2*Pi*sqrt(n)) / (16*n^2). - Vaclav Kotesovec, Sep 08 2015
Convolution inverse of A244276. - Michael Somos, Oct 25 2015
EXAMPLE
G.f. = 1 + 8*x + 39*x^2 + 152*x^3 + 513*x^4 + 1560*x^5 + 4382*x^6 + 11552*x^7 + ...
G.f. = 1/q + 8*q^3 + 39*q^7 + 152*q^11 + 513*q^15 + 1560*q^19 + 4382*q^23 + ...
MATHEMATICA
nmax = 40; CoefficientList[Series[Product[(1 + x^k)^3 / ((1 - x^k)^5 * (1 + x^(2*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] / QPochhammer[ x]^6, {x, 0, n}]; (* Michael Somos, Oct 25 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A)^8 * eta(x^4 + A)^2), n))};
CROSSREFS
Cf. A134414.
Sequence in context: A063002 A055581 A297017 * A097787 A215731 A144414
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 26 2007
STATUS
approved