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A128582
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Expansion of psi(x) * psi(-x^3) / chi(-x^4)^2 in powers of x where psi(), chi() are Ramanujan theta functions.
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4
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1, 1, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 3, 1, 0, 0, 1, 3, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 2, 2, 0, 0, 1, 2, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 2, 0, 0, 0, 3, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 3, 0, 0, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(
q) (A010054), chi(q) (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 q^(-5/6) * eta(q^2)^2 * eta(q^3) * eta(q^8)^2 * eta(q^12) / (eta(q) * eta(q^4)^2 * eta(q^6)) in powers of q.
Euler transform of period 24 sequence [ 1, -1, 0, 1, 1, -1, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, -1, 1, 1, 0, -1, 1, -2, ...].
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 + x^(4*k))^2 * (1 - x^(6*k - 3)) * (1 - x^(12*k)).
A128580(3*n + 2) = -2 * a(n). a(4*n) = A128583. a(4*n + 1) = A128591(n). a(4*n + 2) = a(4*n + 3) = 0.
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EXAMPLE
| 1 + x + x^4 + 2*x^5 + x^8 + x^9 + 2*x^12 + x^13 + x^16 + x^17 + 2*x^20 + ...
q^5 + q^11 + q^29 + 2*q^35 + q^53 + q^59 + 2*q^77 + q^83 + q^101 + q^107 + ...
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PROG
| (PARI) {a(n) = if( n<0, 0, n = 6*n + 5; -1/2 * sumdiv( n, d, kronecker( -12, d) * kronecker( 8, n/d)))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^8 + A)^2 * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)), n))}
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CROSSREFS
| Cf. A128580, A128583, A128591.
Sequence in context: A159708 A144625 A025442 * A101606 A125005 A122179
Adjacent sequences: A128579 A128580 A128581 * A128583 A128584 A128585
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Mar 11 2007
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