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%I #45 Aug 06 2020 00:31:06
%S 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,
%T 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,
%U 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
%N Expansion of Sum_{k>0} (-1)^(k+1) q^(k^2) / ((1-q)(1-q^3)(1-q^5)...(1-q^(2k-1))).
%C |a(n)|<3 if n<1036, a(1036)=3. - _Michael Somos_, Sep 16 2006
%D 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.
%D F. J. Dyson, Selected Papers, Am. Math. Soc., 1996, p. 204.
%H Robert Israel, <a href="/A003475/b003475.txt">Table of n, a(n) for n = 1..2500</a>
%H G. E. Andrews, F. J. Dyson and D. Hickerson, <a href="http://dx.doi.org/10.1007/BF01388778">Partitions and indefinite quadratic forms</a>, Invent. Math. 91 (1988) 391-407.
%H G. E. Andrews, F. G. Garvan, and J. Liang, <a href="http://qseries.org/fgarvan/papers/spt-parity.pdf">Self-conjugate vector partitions and the parity of the spt-function</a>, Acta Arithmetica Vol. 158, Issue 3: 199-218 (2013) doi.org/10.4064/aa158-3-1
%H Alexander E. Patkowski, <a href="http://dx.doi.org/10.1016/j.disc.2009.10.001">A note on the rank parity function</a>, Discrete Math. 310 (2010), 961-965.
%H D. Zagier, <a href="http://people.mpim-bonn.mpg.de/zagier/files/qmf/fulltext.pdf">Quantum modular forms</a>, Example 1 in Quanta of Maths: Conference in honor of Alain Connes, Clay Mathematics Proceedings 11, AMS and Clay Mathematics Institute 2010, 659-675
%F Define c(24*k + 1) = A003406(k), c(24*k - 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
%F G.f.: x + x^2 * (1 - x^2) + x^3 * (1 - x^2) * (1 - x^4) + x^4 * (1 - x^2) * (1 - x^4) * (1 - x^6) + ... . - _Michael Somos_, Aug 18 2006
%e G.f. = x + x^2 + x^3 - x^7 - x^8 - x^10 - x^11 - x^13 + x^15 - x^16 + ...
%p P:=n->mul((1-q^(2*i+1)),i=0..n-1):
%p t5:=add((-1)^(n+1)*q^(n^2)/P(n),n=1..40):
%p t6:=series(t5,q,40); # Based on Patkowski, 2010, Eq. 3.1. - _N. J. A. Sloane_, Jun 29 2011
%t a[ n_] := SeriesCoefficient[ 1 - QHypergeometricPFQ[ {x^2}, {x}, x^2, x], {x, 0, n}]; (* _Michael Somos_, Feb 02 2015 *)
%o (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 */
%o (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 */
%Y Cf. A003406, A053251, A158690.
%K sign
%O 1,70
%A _N. J. A. Sloane_
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