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URL: https://oeis.org/A132218

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A132218
Expansion of psi(-x^3) / phi(-x) in powers of x where psi(), phi() are Ramanujan theta functions.
1
1, 2, 4, 7, 12, 20, 32, 50, 76, 113, 166, 240, 342, 482, 672, 928, 1270, 1724, 2323, 3108, 4132, 5460, 7174, 9376, 12192, 15780, 20332, 26086, 33334, 42432, 53817, 68018, 85680, 107584, 134674, 168092, 209210, 259680, 321484, 396996, 489056, 601052, 737024
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
FORMULA
Expansion of q^(-3/8) * eta(q^2) * eta(q^3) * eta(q^12) / (eta(q)^2 * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 0, ...].
G.f.: Product_{k>0} (1 + x^k) * (1 + x^k + x^(2*k)) * (1 + x^(6*k)).
a(n) ~ 5^(1/4) * exp(sqrt(5*n/6)*Pi) / (2^(11/4) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015
EXAMPLE
G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 20*x^5 + 32*x^6 + 50*x^7 + 76*x^8 + ...
G.f. = q^3 + 2*q^11 + 4*q^19 + 7*q^27 + 12*q^35 + 20*q^43 + 32*q^51 + 50*q^59 + ...
MATHEMATICA
nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(12*k))/( (1-x^k) * (1+x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)
a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/8) EllipticTheta[ 2, Pi/4, x^(3/2)] / EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Nov 01 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^2 * eta(x^6 + A)), n))};
CROSSREFS
Sequence in context: A122515 A193840 A036372 * A101230 A128129 A014968
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 13 2007
STATUS
approved