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A128643
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Expansion of (b(q^2) / b(q))^3 in powers of q where b() is a cubic AGM function.
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4
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1, 9, 45, 171, 549, 1566, 4095, 10008, 23157, 51201, 108918, 224100, 447831, 872118, 1659672, 3093498, 5658453, 10173762, 18006021, 31408092, 54053190, 91869192, 154331028, 256447080, 421789671, 687086127, 1109128014, 1775103507
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of (chi(-q^3) / chi(-q)^3)^3 in powers of q where chi() is a Ramanujan theta function.
Expansion of ((eta(q^2) / eta(q))^3 * (eta(q^3) / eta(q^6)))^3 in powers of q.
Euler transform of period 6 sequence [ 9, 0, 6, 0, 9, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v * (1 - v) * (1 + 8*u) + (u - v)^2.
G.f.: (Product_{k>0} (1 + x^k) / (1 + x^(3*k))^3)^3
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (1 / 8) g(t) where q = exp(2 pi i t) and g() is g.f. for A105559.
a(n) = 9 * A128638(n) unless n = 0. -4 * a(n) = A193522(2*n). Convolution inverse of A128642.
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EXAMPLE
| 1 + 9*q + 45*q^2 + 171*q^3 + 549*q^4 + 1566*q^5 + 4095*q^6 + 10008*q^7 + ...
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MATHEMATICA
| a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3, q^6] / QPochhammer[ q, q^2]^3)^3, {q, 0, n}]
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PROG
| (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x^2 + A) / eta(x + A))^3 * eta(x^3 + A) / eta(x^6 + A))^3, n))}
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CROSSREFS
| Cf. A105559, A128638, A128642, A193522.
Sequence in context: A145458 A145137 A144902 * A036826 A022574 A050574
Adjacent sequences: A128640 A128641 A128642 * A128644 A128645 A128646
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Mar 16 2007
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