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A215598 Expansion of phi(-x^2) * f(x)^3 in powers of x where phi(), f() are Ramanujan theta functions.

%I A215598 #21 Feb 16 2025 08:33:18
%S A215598 1,3,-2,-11,0,10,-7,0,16,6,9,-10,-18,0,-14,11,0,-22,16,-6,0,-3,0,48,
%T A215598 14,0,0,0,-33,-26,30,0,2,-16,0,-10,-13,0,-48,26,0,0,18,0,34,19,30,-16,
%U A215598 0,0,-2,-6,0,22,-34,-21,14,42,0,0,-48,0,0,-80,0,-22,-23,0
%N A215598 Expansion of phi(-x^2) * f(x)^3 in powers of x where phi(), f() are Ramanujan theta functions.
%C A215598 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A215598 G. C. Greubel, <a href="/A215598/b215598.txt">Table of n, a(n) for n = 0..1000</a>
%H A215598 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A215598 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A215598 Expansion of q^(-1/8) * eta(q^2)^11 / (eta(q)^3 * eta(q^4)^4) in powers of q.
%F A215598 Euler transform of period 4 sequence [3, -8, 3, -4, ...].
%F A215598 a(n) = b(8*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - Kronecker(2, p) * p * b(p^(e-2)). b(8*n + 5) = 4 * i * A215596(n).
%e A215598 G.f. = 1 + 3*x - 2*x^2 - 11*x^3 + 10*x^5 - 7*x^6 + 16*x^8 + 6*x^9 + 9*x^10 + ...
%e A215598 G.f. = q + 3*q^9 - 2*q^17 - 11*q^25 + 10*q^41 - 7*q^49 + 16*q^65 + 6*q^73 + 9*q^81 + ...
%o A215598 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^11 / (eta(x + A)^3 * eta(x^4 + A)^4), n))};
%o A215598 (PARI) {a(n) = my(A, p, e, u, v, s, x, y, a0, a1); if( n<0, 0, n = n*8 + 1; A = factor(n); simplify( prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, s = p * kronecker( 2, p); if( p%4==3, if( e%2, 0, (-s)^(e/2)), if( p%8==1, for( y=1, sqrtint(p\16), if( issquare( p - 16*y^2, &u), v=y; if( u%4!=1, u=-u); break)); a0 = 1; a1 = x = 2 * u * (-1)^(u\4 + v)); if( p%8==5, forstep( y=1, sqrtint(p\4), 2, if( issquare( p - 4*y^2, &v), u=y; if( u%4!=1, u=-u); if( v%4!=1, v=-v); break)); a0 = 1; a1 = x = 4 * I * u * (-1)^(v\4)); for( i=2, e, y = x*a1 - s*a0; a0=a1; a1=y); a1)))))};
%Y A215598 Cf. A215596.
%K A215598 sign
%O A215598 0,2
%A A215598 _Michael Somos_, Aug 16 2012