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Search: id:A081362
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| A081362 |
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Expansion of q^(1/24) * eta(q) / eta(q^2) in powers of q. |
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+0
4
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| 1, -1, 0, -1, 1, -1, 1, -1, 2, -2, 2, -2, 3, -3, 3, -4, 5, -5, 5, -6, 7, -8, 8, -9, 11, -12, 12, -14, 16, -17, 18, -20, 23, -25, 26, -29, 33, -35, 37, -41, 46, -49, 52, -57, 63, -68, 72, -78, 87, -93, 98, -107, 117, -125, 133, -144, 157, -168, 178, -192, 209, -223, 236, -255, 276, -294, 312, -335, 361, -385
(list; graph; listen)
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OFFSET
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0,9
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COMMENT
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(Number of partitions of n into an even number of parts) - (number of partitions of n into an odd number of parts). [Fine]
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REFERENCES
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N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 58, Eq. (26.56) [essentially the function phi(q)]; also p. 38, Eq. (22.14).
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FORMULA
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G.f.: Product_{k>0} (1-x^(2k-1)) = Product_{k>0} 1/(1+x^k) = 1+Sum_{k>0} (-x)^k/(Product_{i=1..k} (1-x^i)).
This is the convolution inverse of A000009 (partitions into distinct parts) - i.e. the negation of the INVERTi transform of A000009. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jan 06 2006
Expansion of chi(-q) in powers of q where chi() is a Ramanujan theta function.
Given g.f. A(x), then B(x)=A(x^3)^8/x satisfies 0=f(B(x), B(x^2)) where f(u, v)=u^2*v +16*u -v^2.
G.f. A(x) satisfies A(x^2)=A(x)A(-x).
Euler transform of period 2 sequence [ -1, 0, ...].
Expansion of q^(1/24) f1(t) in powers of q = exp(Pi i t) where f1() is a Weber function.
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EXAMPLE
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q^-1 - q^23 - q^71 + q^95 - q^119 + q^143 - q^167 + 2*q^191 - 2*q^215 + ...
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PROGRAM
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(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)/eta(x^2+A), n))}
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CROSSREFS
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A000700(n)=(-1)^n a(n).
Sequence in context: A035435 A025775 A000700 this_sequence A112216 A058688 A132322
Adjacent sequences: A081359 A081360 A081361 this_sequence A081363 A081364 A081365
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Mar 18 2003
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