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A143063
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Expansion of the product of a false theta function and a Ramanujan theta function in powers of x.
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0
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1, 0, 0, 2, 0, 0, 0, 2, 2, 0, 2, 2, 2, 0, 2, 4, 4, 2, 4, 6, 4, 4, 4, 8, 8, 6, 8, 12, 10, 10, 12, 16, 16, 14, 18, 22, 22, 20, 24, 30, 32, 30, 36, 42, 42, 42, 48, 56, 60, 58, 66, 76, 78, 80, 88, 102, 106, 108, 120, 134, 140, 144, 158, 178, 186, 192, 210, 232, 242, 252, 272, 300
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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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
| G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 235, Entry 9.4.8
S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41
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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
| G.f.: (1 - x + x^2 - x^5 + x^7 - x^12 + x^15 - ...) * (1 + x) * (1 + x^3) * (1 + x^5) * (1 + x^7) * ... [Ramanujan]
G.f.: 1 + 2 * x^3 / (1 - x^4) + 2 * x^8 / ((1 - x^2) * (1 - x^8)) + 2 * x^15 / ((1 - x^2) * (1 - x^4) * (1 - x^12)) + 2 * x^24 / ((1 - x^2) * (1 - x^4) * (1 - x^6) * (1 - x^16)) + ... [Ramanujan]
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EXAMPLE
| 1 + 2*x^3 + 2*x^7 + 2*x^8 + 2*x^10 + 2*x^11 + 2*x^12 + 2*x^14 + 4*x^15 + ...
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PROG
| (PARI) {a(n) = local(m, A); if( n<0, 0, A = x * O(x^n); polcoeff( sum(k=0, n, if( issquare( 24*k + 1, &m), (-1)^(m \ 3) * x^k ), A) / eta(x + A) * eta(x^2 + A)^2 / eta(x^4 + A), n))}
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 + 2 * sum(k=1, sqrtint(n+1) - 1, x^(k^2 + 2*k) / (1 - x^(4*k)) / prod(j=1, k-1, 1 - x^(2*j), 1 + O(x^(n + 1 - k^2 - 2*k)))), n))}
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CROSSREFS
| Cf. a(n) = 2 * A027348(n) unless n=0. Convolution of A143062 and A000700.
Sequence in context: A100699 A108921 A071548 * A175070 A054923 A057108
Adjacent sequences: A143060 A143061 A143062 * A143064 A143065 A143066
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Jul 21 2008
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