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A134077
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Expansion of q^(-1/2) * eta(q)^5 * eta(q^6)^3 / ( eta(q^2) * eta(q^3)^3 ) in powers of q.
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1
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1, -5, 6, 8, -23, 12, 14, -30, 18, 20, -40, 24, 31, -77, 30, 32, -60, 48, 38, -70, 42, 44, -138, 48, 57, -90, 54, 72, -100, 60, 62, -184, 84, 68, -120, 72, 74, -155, 96, 80, -239, 84, 108, -150, 90, 112, -160, 120, 98, -276, 102, 104, -240, 108, 110, -190, 114, 144, -322, 144, 133, -210, 156
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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 psi(q)^4 - 9 * q * psi(q^3)^4 in powers of q where psi() is a Ramanujan theta function.
Expansion of psi(q) * phi(-q)^3 / chi(-q^3)^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/2) * ( b(q)^3 * c(q^2)^2 / ( 3 * c(q) ) )^(1/2) in powers of q where b(), c() are cubic AGM functions.
Euler transform of period 6 sequence [ -5, -4, -2, -4, -5, -4, ...].
a(n) = b(2*n+1) and b(n) is multiplicative with b(2^e) = 0^e, b(3^e) = 4 - 3^(e+1), b(p^e) = (p^(e+1) - 1)/(p - 1) if p>5.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 18 (t/i)^2 g(t) where q = exp(2 pi i t) and g(t) is g.f. for A124449
G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(2*k))^2 * (1 - x^k + x^(2*k))^3.
G.f.: Sum_{k>0} k * f(x^k) - 9 * k * f(x^(3*k)) where f(x) = x * (1 - x) / ( (1 + x) * (1 + x^2) ).
G.f.: f(x) - 3 * f(x^2) - 9 * f(x^3) + 2 * f(x^4) + 27 * f(x^6) - 18 * f(x^12) where f() is g.f. of A000203.
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EXAMPLE
| q - 5*q^3 + 6*q^5 + 8*q^7 - 23*q^9 + 12*q^11 + 14*q^13 - 30*q^15 + ...
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PROG
| (PARI) {a(n) = local(A); if ( n<0, 0, A = x * O(x^n) ; polcoeff( eta(x + A)^5 * eta(x^6 + A)^3 / ( eta(x^2 + A) * eta(x^3 + A)^3 ), n))}
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CROSSREFS
| 6 * A098098(n) = a(3*n+2).
Sequence in context: A054378 A129319 A024570 * A120128 A019149 A172997
Adjacent sequences: A134074 A134075 A134076 * A134078 A134079 A134080
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Oct 06 2007
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