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%I #19 Mar 12 2021 22:24:46
%S 1,-1,0,-1,-3,3,1,3,0,0,-2,0,5,-5,-3,-6,0,0,5,3,0,-1,5,0,-7,10,0,2,1,
%T 0,-7,0,-3,-5,-7,0,1,0,0,7,11,-9,0,-9,0,6,9,0,5,3,9,0,-7,0,0,-10,0,-5,
%U 0,3,-18,2,0,11,0,0,-10,-5,9,7,-14,0,0,0,0,11,9
%N Expansion of psi(-x) * f(-x^4)^3 in powers of x where psi(), f() 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="/A215596/b215596.txt">Table of n, a(n) for n = 0..1000</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 q^(-5/8) * eta(q) * eta(q^4)^4 / eta(q^2) in powers of q.
%F Euler transform of period 4 sequence [-1, 0, -1, -4, ...].
%F a(n) = b(8*n + 5) / (4*i) 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)).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (128 t)) = 2^(9/2) * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A215597.
%e G.f. = 1 - x - x^3 - 3*x^4 + 3*x^5 + x^6 + 3*x^7 - 2*x^10 + 5*x^12 - 5*x^13 + ...
%e G.f. = q^5 - q^13 - q^29 - 3*q^37 + 3*q^45 + q^53 + 3*q^61 - 2*q^85 + 5*q^101 + ...
%t A215596[n_] := SeriesCoefficient[(QPochhammer[x]*QPochhammer[x^4]^4)/ QPochhammer[x^2], {x, 0, n}]; Table[A215596[n], {n, 0, 50}] (* _G. C. Greubel_, Oct 01 2017 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^4 / eta(x^2 + A), n))};
%o (PARI) {a(n) = my(A, p, e, u, v, s, x, y, a0, a1); if( n<0, 0, n = n*8 + 5; A = factor(n); simplify( 1 / (4*I) * 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 Cf. A215597.
%K sign
%O 0,5
%A _Michael Somos_, Aug 16 2012
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