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A106507
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G.f.: Product_{k>0} (1-x^(2k-1))/(1-x^(2k)).
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1
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1, -1, 1, -2, 3, -4, 5, -7, 10, -13, 16, -21, 28, -35, 43, -55, 70, -86, 105, -130, 161, -196, 236, -287, 350, -420, 501, -602, 722, -858, 1016, -1206, 1431, -1687, 1981, -2331, 2741, -3206, 3740, -4368, 5096, -5922, 6868, -7967, 9233, -10670, 12306, -14193, 16357, -18803, 21581
(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).
Expansion of 1/psi(q) in powers of q where psi() is a Ramanujan theta function.
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REFERENCES
| S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41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
C. Adiga, N. Anitha, T. Kim, Transformations of Ramanujan's Summation Formula and its Applications, See page 5
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FORMULA
| Euler transform of period 2 sequence [ -1, 1, ...].
Given g.f. A(x), then B(x)=A(x^8)/x satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=u^4(w^4 +4v^4) -v^6w^2.
Given g.f. A(x), then B(x)=A(x^8)/x satisfies 0=f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6)=u1*u2*u6^3 +u2^2*u3^3 -u3^3*u6^2.
Given g.f. A(x), then B(x)=A(x^8)/x satisfies 0=f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6)=u1^3*u6^2 +3*u1^3*u2^2 -u2^3*u3*u6.
Sum_{k>=0} a(k)x^(8k-1) = 1/(Sum_{k} x^((4k+1)^2)).
Expansion of q^(1/8)* eta(q)/eta(q^2)^2 in powers of q.
G.f.: 1 / (1 + x + x^3 + x^6 + ...) = 1 - x * (1 - x) / (1 - x^2)^2 + x^4 * (1 - x) * (1 - x^2) / ((1 - x^2)^2 * (1 - x^4)^2) + ... [Ramanujan] (Michael Somos, Jul 21 2008)
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(1/2) (t / i)^(-1/2) g(t) where q = exp(2 pi i t) and g() is g.f. for A015128. - Michael Somos Nov 01 2008
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EXAMPLE
| 1/q - q^7 + q^15 - 2*q^23 + 3*q^31 - 4*q^39 + 5*q^47 - 7*q^55 +...
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PROG
| (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)/eta(x^2+A)^2, n))}
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CROSSREFS
| Convolution inverse of A010054. (-1)^n * A006950(n) = a(n).
Sequence in context: A014670 A036034 A006950 * A052335 A160333 A174578
Adjacent sequences: A106504 A106505 A106506 * A106508 A106509 A106510
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KEYWORD
| sign
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AUTHOR
| Michael Somos, May 04 2005
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EXTENSIONS
| Definition changed by N. J. A. Sloane (njas(AT)research.att.com), Aug 14 2007
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