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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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%I #27 Mar 12 2021 22:24:46

%S 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,

%T -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,

%U 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

%N 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.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A187097/b187097.txt">Table of n, a(n) for n = 0..2500</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F 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

%F 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.

%F 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).

%F Convolution square is A187096.

%e G.f. = 1 - x^2 - x^3 + x^4 - x^5 - x^8 + x^9 + 2*x^11 + x^17 - x^21 - x^22 + ...

%e G.f. = q - q^5 - q^7 + q^9 - q^11 - q^17 + q^19 + 2*q^23 + q^35 - q^43 + ...

%t 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 *)

%t 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 *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( sqrt( Ser( ellan( ellinit([ 0, 1, 1, 1, 0], 1), n+1))), n))};

%o (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]))))};

%o (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))};

%Y Cf. A187096.

%K sign

%O 0,12

%A _Michael Somos_, Mar 04 2011