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A136028
Expansion of (phi(q) * phi(q^2))^3 in powers of q where phi() is a Ramanujan theta function.
2
1, 6, 18, 44, 90, 144, 212, 288, 330, 418, 528, 588, 836, 1008, 1056, 1440, 1386, 1356, 1894, 1644, 2064, 2880, 2484, 3168, 3428, 2838, 3696, 3864, 4128, 5040, 5280, 5760, 5418, 5656, 5988, 5376, 7678, 8208, 7572, 10080, 8208, 7788, 10560, 8652, 10404, 13104
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 (eta(q^2) * eta(q^4))^9 / (eta(q) * eta(q^8))^6 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1/(8 t)) = 2^(9/2) (t/i)^3 f(t) where q = exp(2 Pi i t).
G.f.: (Product_{k>0} (1 - x^k)^2 * (1 + x^k)^4 * (1 + x^(2*k)) / (1 + x^(4*k))^2)^3.
a(n) = A029713(n) + 6 * A030207(n). Convolution of A033715 and A097057.
a(n) = A028578(4*n). - Michael Somos, Oct 14 2015
EXAMPLE
G.f. = 1 + 6*q + 18*q^2 + 44*q^3 + 90*q^4 + 144*q^5 + 212*q^6 + 288*q^7 + ...
MATHEMATICA
nmax=60; CoefficientList[Series[Product[((1-x^k)^2 * (1+x^k)^4 * (1+x^(2*k)) / (1+x^(4*k))^2)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2])^3, {q, 0, n}]; (* Michael Somos, Oct 14 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x^2 + A) * eta(x^4 + A))^3 / (eta(x + A) * eta(x^8 + A))^2)^3, n))};
(Magma) A := Basis( ModularForms( Gamma1(8), 3), 46); A[1] + 6*A[2] + 18*A[3] + 44*A[4] + 90*A[5] + 144*A[6] + 212*A[7]; /* Michael Somos, Oct 14 2015 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Dec 10 2007
STATUS
approved