A003475 Expansion of Sum_{k>0} (-1)^(k+1) q^(k^2) / ((1-q)(1-q^3)(1-q^5)...(1-q^(2k-1))).
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
Keywords
Examples
G.f. = x + x^2 + x^3 - x^7 - x^8 - x^10 - x^11 - x^13 + x^15 - x^16 + ...
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
- Robert Israel, Table of n, a(n) for n = 1..2500
- 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
Programs
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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
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Mathematica
a[ n_] := SeriesCoefficient[ 1 - QHypergeometricPFQ[ {x^2}, {x}, x^2, x], {x, 0, n}]; (* Michael Somos, Feb 02 2015 *) -
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 */
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
Comments