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A204770
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Expansion of psi(x^3) * f(-x) in powers of x where psi(), f() are Ramanujan theta functions.
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2
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1, -1, -1, 1, -1, 0, 0, 1, 1, 1, 0, -1, -1, 0, 1, -2, 1, 0, 0, -1, -1, -1, 1, 1, -1, 2, 1, 0, 0, 1, 0, 0, -1, -1, 0, 1, 0, 1, -1, 0, 0, 0, -1, -1, 0, 0, -1, -1, 0, -1, 1, 1, 2, -1, 1, 0, 1, 0, -1, 0, 1, 0, 0, 1, -1, -2, 1, 1, 1, 1, -1, 1, 0, -1, 0, 0, 0, -1, -1, -1, -2, 1, 0, 0, 1, -1, -2, 1, -1, 2, 0, 1, 1, 0, 0, 0, 0, 0, -1, -1, 0, 1, 1, 0, 0
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OFFSET
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0,16
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COMMENTS
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Conjecture: density of zero terms slowly approaches 1.
The absolute value of the terms grows slowly. The first two with abs(a(n)) = 3 or 4 are a(570) = 3. a(940) = -4.
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LINKS
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FORMULA
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Expansion of q^(-5/12) * eta(q) * eta(q^6)^2 / eta(q^3) in powers of q.
Euler transform of period 6 sequence [-1, -1, 0, -1, -1, -2, ...].
G.f.: Product_{k > 0} (1 - x^k) * (1 - x^(6*k)) / (1 - x^(6*k - 3)).
G.f.: Product_{k > 0} (1 - x^k) * (1 + x^(3*k)) * (1 - x^(6*k)). - Michael Somos, May 24 2015
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EXAMPLE
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G.f. = 1 - x - x^2 + x^3 - x^4 + x^7 + x^8 + x^9 - x^11 - x^12 + x^14 - 2*x^15 + ...
G.f. = q^5 - q^17 - q^29 + q^41 - q^53 + q^89 + q^101 + q^113 - q^137 - q^149 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^6]^2 / QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, May 24 2015 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^2 / eta(x^3 + A), n))};
(PARI) {a(n) = my(A, p, e, x, w=quadgen(-8)); if( n<0, 0, n = 12*n + 5; A = factor(n); simplify( 1/w * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 0, p%12 == 11, !(e%2), p%12 == 5, forstep(y = 3, sqrtint(2*p), 6, if( issquare(2*p - y^2, &x), if( x%6==5, x=-x); x = (x-1)/6; break)); (-1)^(e*x) * [1, w, -1, 0, -1, -w, 1, 0][e%8+1], x=0; forstep(y = 0, sqrtint(p), 6, if( issquare(p - y^2, &x), if( x%6==5, x=-x); x = [x-1, y]/6; break)); if( x==0, (-1)^(e\2) * !(e%2), (-1)^(e*(x[1] + x[2])) * (e+1))))))}; /* Michael Somos, May 24 2015 */
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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