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A226289
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Expansion of f(-x) * phi(x^3) in powers of x where f(), phi() are Ramanujan theta functions.
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5
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1, -1, -1, 2, -2, -1, 0, 1, 2, 0, 2, 0, 1, -2, -2, -3, 0, 2, -2, 2, 0, 0, 1, 0, -2, 2, 1, 0, -2, 0, 0, 0, 2, 0, 4, -1, 0, 0, 0, -2, -1, 0, -2, -2, 0, 0, 0, -2, 2, 0, -2, 1, -2, 4, 2, 2, 0, 1, 0, 0, 0, 0, -2, 0, 0, 0, 0, -2, 0, 2, 1, 0, 0, -2, 2, 2, -2, -3, 2
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OFFSET
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0,4
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COMMENTS
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LINKS
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FORMULA
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Expansion of q^(-1/24) * eta(q) * eta(q^6)^5 / (eta(q^3)^2 * eta(q^12)^2) in powers of q.
Euler transform of period 12 sequence [ -1, -1, 1, -1, -1, -4, -1, -1, 1, -1, -1, -2, ...].
a(n) = b(24*n + 1) where b() is multiplicative with b(2^e) = 0, b(3^e) = I^e, b(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 23 (mod 24), b(p^e) = I^e * (1 + (-1)^e) / 2 if p == 5, 7, 11 (mod 24), b(p^e) = (e+1) * s^e where s = Kronecker( 12, x) if p == 1 (mod 24), b(p^e) = (e+1) * s^e where s = -I * Kronecker( 12, x) if p == 19 (mod 24), where p = x^2 + 18 * y^2.
Given the 3-section A = A0 + A1 + A2, then 0 = A0 * A1^2 + A1 * A2^2 + A2 * A0^2.
G.f.: (Product_{k>0} 1 - x^k) * (Sum_{k in Z} x^(3*k^2)).
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EXAMPLE
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G.f. = 1 - x - x^2 + 2*x^3 - 2*x^4 - x^5 + x^7 + 2*x^8 + 2*x^10 + x^12 - 2*x^13 + ...
G.f. = q - q^25 - q^49 + 2*q^73 - 2*q^97 - q^121 + q^169 + 2*q^193 + 2*q^241 + ...
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MATHEMATICA
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a[ n_] := Module[ {m = 24 n + 1, i}, If[ n < 0, 0, Sum[ i = Sqrt[ m - 72 j^2]; If[ IntegerQ @ i, (1 + Boole[ j > 0]) JacobiSymbol[ 12, i], 0], {j, 0, Sqrt[ m / 72]}]]];
a[ n_] := SeriesCoefficient[ QPochhammer[ q] EllipticTheta[ 3, 0, q^3], {q, 0, n}];
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PROG
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(PARI) {a(n) = my(m, i); if( n<0, 0, m = 24*n + 1; sum( j=0, sqrtint( m \ 72), if( issquare( m - 72 * j^2, &i), (1 + (j>0)) * kronecker( 12, i), 0)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^5 / (eta(x^3 + A)^2 * eta(x^12 + A)^2), n))};
(PARI) {a(n) = my(A, p, e, i); 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]; I^e * if( p<5, p>2, if( p%24 == 1 || p%24 == 19, forstep( j = 3 * (p%24>1), sqrtint( p \ 2), 6, if( issquare( p - 2*j^2, &i), break)); (- kronecker( 12, i) * if( p%24>1, 1, I))^e * (e+1), if( e%2, 0, I^((p%24>12)*e)))))))};
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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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