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A137829
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Expansion of psi(q^2) / f(-q)^2 in powers of q where psi(), f() are Ramanujan theta functions.
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3
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1, 2, 6, 12, 25, 46, 86, 148, 255, 420, 686, 1088, 1712, 2634, 4020, 6036, 8988, 13214, 19282, 27840, 39923, 56750, 80160, 112384, 156660, 216958, 298894, 409420, 558119, 756950, 1022090, 1373760, 1838932, 2451366, 3255480, 4306920, 5678104
(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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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^(-1/6) * eta(q^4)^2 / (eta(q)^2 * eta(q^2)) in powers of q.
Euler transform of period 4 sequence [ 2, 3, 2, 1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 96^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A137830.
G.f.: Product_{k>0} (1 + x^k) * (1 + x^(2*k))^2 / (1 - x^k).
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EXAMPLE
| q + 2*q^7 + 6*q^13 + 12*q^19 + 25*q^25 + 46*q^31 + 86*q^37 + 148*q^43 + ...
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PROG
| (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 / eta(x + A)^2 / eta(x^2 + A), n))}
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CROSSREFS
| A137828(4*n+1) = 2 * a(n). Half of A201078, which gives another application.
Sequence in context: A180071 A034882 A175943 * A045925 A128020 A116562
Adjacent sequences: A137826 A137827 A137828 * A137830 A137831 A137832
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Feb 12 2008
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