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A261156
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Expansion of chi(q) * chi(-q^9) / (chi(-q) * chi(q^9)) in powers of q where chi() is a Ramanujan theta function.
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2
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1, 2, 2, 4, 6, 8, 12, 16, 22, 28, 36, 48, 60, 76, 96, 120, 150, 184, 228, 280, 340, 416, 504, 608, 732, 878, 1052, 1252, 1488, 1768, 2088, 2464, 2902, 3408, 3996, 4672, 5460, 6364, 7400, 8600, 9972, 11544, 13344, 15400, 17752, 20424, 23472, 26944, 30876, 35346
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OFFSET
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0,2
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COMMENTS
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Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
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Expansion of eta(q^2)^3 * eta(q^9)^2 * eta(q^36) / (eta(q)^2 * eta(q^4) * eta(q^18)^3) in powers of q.
G.f. A(x) = B(x) / B(x^9) where B(x) is the g.f. of A080054.
Euler transform of period 36 sequence [ 2, -1, 2, 0, 2, -1, 2, 0, 0, -1, 2, 0, 2, -1, 2, 0, 2, 0, 2, 0, 2, -1, 2, 0, 2, -1, 0, 0, 2, -1, 2, 0, 2, -1, 2, 0, ...].
a(n) = 2 * A233693(n) unless n=0. a(2*n) = 2 * A123629(n) = 2 * A212484(n) unless n=0.
a(3*n) = A186924(n). a(3*n) = 4 * A187100(n) unless n=0.
a(n) = (-1)^n * A260215(n). - Michael Somos, Aug 14 2015
a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017
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EXAMPLE
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G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 12*x^6 + 16*x^7 + 22*x^8 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2] QPochhammer[ -q, q] QPochhammer[ q^9, q^18] QPochhammer[ q^9, -q^9], {q, 0, n}];
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^9 + A)^2 * eta(x^36 + A) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^18 + A)^3), n))};
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CROSSREFS
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Cf. A080054, A123629, A186924, A187100, A212484, A233693, A260215.
Sequence in context: A051466 A320193 A260215 * A080015 A210030 A080054
Adjacent sequences: A261153 A261154 A261155 * A261157 A261158 A261159
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KEYWORD
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nonn
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AUTHOR
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Michael Somos, Aug 10 2015
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STATUS
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approved
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