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A003475
Expansion of Sum_{k>0} (-1)^(k+1) q^(k^2) / ((1-q)(1-q^3)(1-q^5)...(1-q^(2k-1))).
7
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
OFFSET
1,70
COMMENTS
|a(n)|<3 if n<1036, a(1036)=3. - Michael Somos, Sep 16 2006
REFERENCES
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.
LINKS
G. E. Andrews, F. J. Dyson and D. Hickerson, Partitions and indefinite quadratic forms, Invent. Math. 91 (1988) 391-407.
G. E. Andrews, F. G. Garvan, and J. Liang, Self-conjugate vector partitions and the parity of the spt-function, Acta Arithmetica Vol. 158, Issue 3: 199-218 (2013) doi.org/10.4064/aa158-3-1
Alexander E. Patkowski, A note on the rank parity function, Discrete Math. 310 (2010), 961-965.
D. Zagier, Quantum modular forms, Example 1 in Quanta of Maths: Conference in honor of Alain Connes, Clay Mathematics Proceedings 11, AMS and Clay Mathematics Institute 2010, 659-675
FORMULA
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
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
EXAMPLE
G.f. = x + x^2 + x^3 - x^7 - x^8 - x^10 - x^11 - x^13 + x^15 - x^16 + ...
MAPLE
P:=n->mul((1-q^(2*i+1)), i=0..n-1):
t5:=add((-1)^(n+1)*q^(n^2)/P(n), n=1..40):
t6:=series(t5, q, 40); # Based on Patkowski, 2010, Eq. 3.1. - N. J. A. Sloane, Jun 29 2011
MATHEMATICA
a[ n_] := SeriesCoefficient[ 1 - QHypergeometricPFQ[ {x^2}, {x}, x^2, x], {x, 0, n}]; (* Michael Somos, Feb 02 2015 *)
PROG
(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 */
CROSSREFS
KEYWORD
sign
STATUS
approved