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A187097
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Expansion of psi(x^4) * phi(x^38) - x^2 * psi(x) * psi(x^19) + x^9 * phi(x^2) * psi(x^76) in powers of x where phi(), psi() are Ramanujan theta functions.
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2
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1, 0, -1, -1, 1, -1, 0, 0, -1, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, -1, -1, 0, 0, 0, 1, 0, 0, -1, -1, 0, 0, 0, 0, -1, 0, 1, 0, 1, 2, 1, 0, 0, 0, 0, -1, 0, -1, 2, 0, 0, 0, 0, 0, 0, -2, 0, 1, 0, 0, 1, 0, 0, -1, -1, 0, -1, -1, 0, 0, 0, 0, -1, 0, -1, 0, 2, 0, -2, 2, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 2, -1, 0, 0
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OFFSET
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0,12
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COMMENTS
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LINKS
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FORMULA
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Expansion of f(-x^2, -x^3) * f(-x^38, -x^57) + x^4 * f(-x, -x^4) * f(-x^19, -x^76) = f(-x^2, -x^5) * f(-x^57, -x^76) - x^3 * f(-x, -x^7) * f(-x^38, -x^95) - x^8 * f(-x^3, -x^4) * f(-x^19, -x^114) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Jan 06 2013
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(19^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if -1 = Kronecker(-19, p), b(p^e) = e + 1 if p = x^2 + 19 * y^2, b(p^e) = Kronecker(-3, e+1) if p = 4*x^2 + 2*x*y + 5*y^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (76 t)) = 76^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
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EXAMPLE
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G.f. = 1 - x^2 - x^3 + x^4 - x^5 - x^8 + x^9 + 2*x^11 + x^17 - x^21 - x^22 + ...
G.f. = q - q^5 - q^7 + q^9 - q^11 - q^17 + q^19 + 2*q^23 + q^35 - q^43 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (-EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^19] + 2 (EllipticTheta[ 3, 0, q^76] EllipticTheta[ 2, 0, q^4] + EllipticTheta[ 3, 0, q^4] EllipticTheta[ 2, 0, q^76])) / 4, {q, 0, 2 n + 1}]; (* Michael Somos, Oct 17 2016 *)
a:= CoefficientList[Series[q^(-1/2) (2 EllipticTheta[2, 0, q^2] EllipticTheta[3, 0, q^38] - EllipticTheta[2, 0, q^(1/2)] EllipticTheta[2, 0, q^(19/2)] + 2 EllipticTheta[3, 0, q^2] EllipticTheta[2, 0, q^38])/4, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Aug 14 2018 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( sqrt( Ser( ellan( ellinit([ 0, 1, 1, 1, 0], 1), n+1))), n))};
(PARI) {a(n) = my(A, p, e, y); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p==19, 1, -1 == kronecker( -19, p), !(e%2), y = 0; for( i=1, sqrtint(p\19), if( issquare(p - 19*i^2, &y), break)); if( y, e+1, [ 1, -1, 0][e%3 + 1]))))};
(PARI) {a(n) = my(phi1, psi1); if( n<0, 0, phi1 = 1 + 2 * sum( k=1, sqrtint( n), x^k^2, x * O(x^n)); psi1 = sum( k=1, ( sqrtint( 8*n + 1) + 1 ) \ 2, x^((k^2 - k)/2), x * O(x^n)); polcoeff( subst( psi1 + x * O(x^(n \ 4)), x, x^4) * subst( phi1 + x * O(x^(n \ 38)), x, x^38) - x^2 * psi1 * subst( psi1 + x * O(x^(n \ 19)), x, x^19) + x^9 * subst( phi1 + x * O(x^(n \ 2)), x, x^2) * subst( psi1 + x * O(x^(n \ 76)), x, x^76), n))};
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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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