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Search: id:A003475
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| A003475 |
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Expansion of Sum { (-1)^n q^(n^2) / ((1-q)(1-q^3)(1-q^5)...(1-q^(2n-1))) }; n=0..inf. |
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+0
3
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| 1, 1, 1, 0, 0, 0, -1, -1, 0, -1, -1, 0, -1, 0, 1, -1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, -1, 0, 0, 0, 1, -1, -1, -1, 0, 0, -1, 0, -1, 0, 0, -1, -1, -1, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 1, 0, 1, 1, 0, 1, -1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1, 0, 0, 0, -1, -2, 0, 0, -1, 0, 1, 0, -1, -1, -1, 0, 0, 0
(list; graph; listen)
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OFFSET
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1,70
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COMMENT
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|a(n)|<3 if n<1036, a(1036)=3. - Michael Somos Sep 16 2006
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REFERENCES
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F. J. Dyson, A walk through Ramanujan's garden, pp. 7-28 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
F. J. Dyson, Selected Papers, Am. Math. Soc., 1996, p. 204.
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FORMULA
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Define c(24k+1)=A003406(k), c(24k-1)=-2*A003475(k), c(n)=0 otherwise. Then c(n) is multiplicative with c(2^e)=c(3^e)=0^e, c(p^e) = (-1)^(e/2)*(1+(-1)^e)/2 if p == 7, 17 (mod 24), c(p^e) = (1+(-1)^e)/2 if p == 5, 11, 13, 19 (mod 24), c(p^e) = (e+1)*(-1)^(y*e) where p == 1, 23 (mod 24) and p = x^2-72*y^2 . - Michael Somos Aug 17 2006 */
G.f.: x +x^2(1-x^2) +x^3(1-x^2)(1-x^4) +x^4(1-x^2)(1-x^2)(1-x^6) +... . - Michael Somos Aug 18 2006
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PROGRAM
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(PARI) {a(n)=local(A, p, e, x, y); if(n<0, 0, n=24*n-1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p<5, 0, if(p%24>1&p%24<23, if(e%2, 0, if(p%24==7|p%24==17, (-1)^(e/2), 1)), x=y=0; if(p%24==1, forstep(i=1, sqrtint(p), 2, if(issquare((i^2+p)/2, &y), x=i; break)), for(i=1, sqrtint(p\2), if(issquare(2*i^2+p, &x), y=i; break))); (e+1)*(-1)^( (x+if((x-y)%6, y, -y))/6*e)))))/-2)} /* Michael Somos Aug 17 2006 */
(PARI) {a(n)=local(A); if(n<1, 0, A=-1+x*O(x^n); polcoeff( sum(k=1, sqrtint(n), A*= 1/(1-x^(1-2*k))*(1+x*O(x^(n-k^2)))), n))} /* Michael Somos Sep 16 2006 */
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CROSSREFS
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Cf. A053251.
Sequence in context: A085491 A116681 A131371 this_sequence A135767 A070203 A070201
Adjacent sequences: A003472 A003473 A003474 this_sequence A003476 A003477 A003478
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KEYWORD
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sign
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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