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A129402
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Expansion of phi(x^3) * psi(x^4) + x * phi(x)* psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions.
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2
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1, 1, 2, 2, 1, 2, 0, 2, 0, 0, 2, 0, 3, 1, 2, 2, 2, 4, 0, 0, 0, 0, 2, 0, 3, 0, 2, 4, 0, 2, 0, 2, 0, 0, 0, 0, 2, 3, 4, 2, 1, 2, 0, 2, 0, 0, 2, 0, 2, 2, 2, 2, 4, 2, 0, 0, 0, 0, 0, 0, 3, 0, 4, 2, 0, 2, 0, 2, 0, 0, 0, 0, 4, 3, 2, 2, 0, 4, 0, 2, 0, 0, 4, 0, 1, 0, 2, 6, 2, 2, 0, 0, 0, 0, 2, 0, 2, 0, 2, 2, 0, 4, 0, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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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REFERENCES
| N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 83, Eq. (32.57).
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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 f(x^2) * f(-x^3) / (chi(-x) * chi(x^6)) in powers of x where f(), chi() are Ramanujan theta functions.
Expansion of q^(-1/2) * eta(q^3) * eta(x^4)^3 * eta(q^6) * eta(q^24) / (eta(q) * eta(q^8) * eta(q^12)^12) in powers of q.
Euler transform of period 24 sequence [ 1, 1, 0, -2, 1, -1, 1, -1, 0, 1, 1, -2, 1, 1, 0, -1, 1, -1, 1, -2, 0, 1, 1, -2, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1, 5, 7, 11 (mod 24), b(p^e) = (1+(-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
a(12*n + 6) = a(12*n + 8) = a(12*n + 9) = a(12*n + 11) = 0. a(3*n + 1) = a(n).
A000377(2*n + 1) = a(n). A128582(n) = a(3*n + 2) / 2. A113780(n) = a(12*n).
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EXAMPLE
| 1 + x + 2*x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^7 + 2*x^10 + 3*x^12 + x^13 + 2*x^14 + ...
q + q^3 + 2*q^5 + 2*q^7 + q^9 + 2*q^11 + 2*q^15 + 2*q^21 + 3*q^25 + q^27 + ...
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PROG
| (PARI) {a(n) = if( n<0, 0, n = 2*n+1; sumdiv( n, d, kronecker( -6, d)))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^4 + A)^3 * eta(x^6 + A) * eta(x^24 + A) / (eta(x + A) * eta(x^8 + A) * eta(x^12 + A)^2), n))}
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CROSSREFS
| Cf. A000377, A128582, A113780.
Sequence in context: A128580 * A134177 A190615 A104405 A156381 A089077
Adjacent sequences: A129399 A129400 A129401 * A129403 A129404 A129405
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Apr 13 2007
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