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A133098
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Expansion of q^(-1) * chi(-q)^2 * chi(-q^15)^2 / ( chi(-q^3) * chi(-q^5) ) in powers of q where chi() is a Ramanujan theta function.
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2
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1, -2, 1, -1, 2, -2, 2, -3, 5, -5, 5, -7, 9, -10, 11, -14, 18, -20, 22, -27, 32, -36, 40, -48, 57, -63, 70, -82, 95, -106, 119, -137, 158, -175, 195, -222, 252, -280, 311, -352, 397, -439, 486, -546, 611, -676, 747, -834, 929, -1024, 1128, -1253, 1389, -1528, 1679, -1857, 2052, -2250, 2467, -2718
(list; graph; refs; listen; history; internal format)
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OFFSET
| -1,2
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (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 eta(q)^2 * eta(q^6) * eta(q^10) * eta(q^15)^2 / ( eta(q^2)^2 * eta(q^3) * eta(q^5) * eta(q^30)^2 ) in powers of q.
Euler transform of period 30 sequence [ -2, 0, -1, 0, -1, 0, -2, 0, -1, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -1, 0, -2, 0, -1, 0, -1, 0, -2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * v - v^2 + 2 * u + 4 * u * v.
G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = 2 / f(t) where q = exp(2 pi i t).
G.f.: (1/x) * Product_{k>0} (1 + x^(3*k)) * (1 + x^(5*k)) / ((1 + x^k)^2 * (1 + x^(15*k))^2 ).
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EXAMPLE
| 1/q - 2 + q - q^2 + 2*q^3 - 2*q^4 + 2*q^5 - 3*q^6 + 5*q^7 - 5*q^8 + ...
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PROG
| (PARI) {a(n) = local(A); if(n<-1, 0, n++; A = x*O(x^n); polcoeff( eta(x+A)^2 * eta(x^6+A) * eta(x^10+A) * eta(x^15+A)^2 / eta(x^2+A)^2 / eta(x^3+A) / eta(x^5+A) / eta(x^30+A)^2, n))}
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CROSSREFS
| A058618(n) = a(n) unless n = 0. Convolution inverse of A123630.
Sequence in context: A161040 A180234 A131059 * A145788 A117592 A117942
Adjacent sequences: A133095 A133096 A133097 * A133099 A133100 A133101
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Sep 10 2007
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