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A134461
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Expansion of (phi(q) * psi(-q))^4 in powers of q where phi(), psi() are Ramanujan theta functions.
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1, 4, -2, -24, -11, 44, 22, -8, 50, -44, -96, 56, -121, -152, 198, 160, 176, 48, -162, 88, -198, -52, 22, -528, 233, 200, -242, -88, -176, 668, 550, 264, -44, -188, 224, -728, 154, -484, -1056, 656, -311, -236, -100, 792, 714, -528, 640, 88, -478, -484, 1566, 968, 192, 780, -1994, -648, -942
(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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REFERENCES
| Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
W. Stein, Modular Forms Database.
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FORMULA
| Expansion of q^(-1/2) * (eta(q^2)^4 / (eta(q) * eta(q^4)))^4 in powers of q.
Euler transform of period 4 sequence [ 4, -12, 4, -8, ...].
a(n) = b(2*n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = b(p)*b(p^(e-1)) -p^3*b(p^(e-2)).
G.f. is Fourier series of a weight 4 level 16 modular form. f(-1/ (16 t)) = 256 (t/i)^4 f(t) where q = exp(2 pi i t).
G.f.: (Product_{k>0} (1 + x^k) * (1 - x^(2*k))^2 / (1 + x^(2*k)))^4.
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EXAMPLE
| q + 4*q^3 - 2*q^5 - 24*q^7 - 11*q^9 + 44*q^11 + 22*q^13 - 8*q^15 + ...
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PROG
| (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x^2 + A)^4 / eta(x + A) / eta(x^4 + A) )^4, n))}
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CROSSREFS
| (-1)^n * A030211 = a(n).
Sequence in context: A157407 A100400 A030211 * A058167 A140331 A095896
Adjacent sequences: A134458 A134459 A134460 * A134462 A134463 A134464
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Oct 26 2007
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