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A111661
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Expansion of eta(q)^4 * eta(q^2) * eta(q^6)^5 / eta(q^3)^4 in powers of q.
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1, -4, 1, 16, -24, -4, 50, -64, 1, 96, -120, 16, 170, -200, -24, 256, -288, -4, 362, -384, 50, 480, -528, -64, 601, -680, 1, 800, -840, 96, 962, -1024, -120, 1152, -1200, 16, 1370, -1448, 170, 1536, -1680, -200, 1850, -1920, -24, 2112, -2208, 256, 2451, -2404, -288, 2720, -2808, -4
(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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REFERENCES
| B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 226 Entry 4(i).
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LINKS
| Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Euler transform of period 6 sequence [ -4, -5, 0, -5, -4, -6, ...].
Expansion of q * psi(q)^2 * phi(-q)^3 * psi(q^3)^2 / phi(-q^3) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos Mar 01 2011
Expansion of (b(q^2)^3 - b(q)^3) / 9 in powers of q where b() is a cubic AGM function. - Michael Somos Mar 01 2011
a(n) is multiplicative with a(2^e) = (-4)^e, a(3^e) = 1, a(p^e) = ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1 (mod 6), a(p^e) = (1 - (-p^2)^(e+1)) / (p^2 + 1) if p == 5 (mod 6). - Michael Somos Mar 01 2011
G.f.: Sum_{k>0} kronecker(k, 3) * k^2 * x^k / (1 - x^(2*k)) = x * Product_{k>0} (1 - x^k)^4 * (1 - x^(2*k)) * (1 + x^(3*k))^5 * (1 - x^(3*k)).
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EXAMPLE
| q - 4*q^2 + q^3 + 16*q^4 - 24*q^5 - 4*q^6 + 50*q^7 - 64*q^8 + q^9 + ...
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PROG
| (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^2 * kronecker(d, 3)))}
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^2 + A) * eta(x^6 + A)^5 / eta(x^3 + A)^4, n))}
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CROSSREFS
| Sequence in context: A099394 A059991 A002568 * A072651 A093035 A126791
Adjacent sequences: A111658 A111659 A111660 * A111662 A111663 A111664
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KEYWORD
| sign,mult
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AUTHOR
| Michael Somos, Aug 08 2005
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