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Search: id:A113652
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| A113652 |
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Expansion of (1-theta_4(q)^2)/4 in powers of q. |
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3
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| 1, -1, 0, -1, 2, 0, 0, -1, 1, -2, 0, 0, 2, 0, 0, -1, 2, -1, 0, -2, 0, 0, 0, 0, 3, -2, 0, 0, 2, 0, 0, -1, 0, -2, 0, -1, 2, 0, 0, -2, 2, 0, 0, 0, 2, 0, 0, 0, 1, -3, 0, -2, 2, 0, 0, 0, 0, -2, 0, 0, 2, 0, 0, -1, 4, 0, 0, -2, 0, 0, 0, -1, 2, -2, 0, 0, 0, 0, 0, -2, 1, -2, 0, 0, 4, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 2, -1, 0, -3, 2, 0, 0, -2, 0
(list; graph; listen)
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OFFSET
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1,5
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REFERENCES
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B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 114 Entry 8(v).
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 576.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987.
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 28, Article 269.
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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a(n) is multiplicative with a(2^e) = -1 if e>0, a(p^e) = e+1 if p == 1 (mod 4), (1+(-1)^e)/2 if p == 3 (mod 4).
Expansion of (1-eta(q)^4/eta(q^2)^2)/4 in powers of q.
Moebius transform is period 8 sequence [1, -2, -1, 0, 1, 2, -1, 0, ...].
G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u2-2*u3+u6-u1^2+3*u3^2+2*u1*u3-4*u2*u6
G.f.: Sum_{k>0} -(-1)^k x^((k^2+k)/2)/(1+x^k) = Sum_{k>0} -(-1)^k x^k/(1+x^(2k)) = Sum_{k>=0} (-1)^k x^(2k+1)/(1+x^(2k+1)).
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EXAMPLE
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q - q^2 - q^4 + 2*q^5 - q^8 + q^9 - 2*q^10 + 2*q^13 - q^16 + 2*q^17 + ...
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PROGRAM
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(PARI) {a(n)=if(n<1, 0, -(-1)^n*sumdiv(n, d, kronecker(-4, d)))}
(PARI) {a(n)=local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, -1, if(p%4==1, e+1, !(e%2))))))}
(PARI) {a(n)=if(n<1, 0, direuler(p=2, n, if(p==2, 1-X/(1-X), 1/(1-X)/(1-kronecker(-4, p)*X)) )[n])}
(PARI) {a(n)=local(A); if(n<1, 0, A=x*O(x^n); polcoeff( (1-eta(x+A)^4/eta(x^2+A)^2)/4, n))}
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CROSSREFS
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Cf. a(n)=-(-1)^n A002654(n). a(n)=-A104794(n)/4 if n>0.
A008441(n)=a(4n+1).
Sequence in context: A151899 A079632 A002654 this_sequence A106139 A052154 A039977
Adjacent sequences: A113649 A113650 A113651 this_sequence A113653 A113654 A113655
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KEYWORD
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sign,mult
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AUTHOR
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Michael Somos, Nov 03 2005
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