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A215596
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Expansion of psi(-x) * f(-x^4)^3 in powers of x where psi(), f() are Ramanujan theta functions.
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3
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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, 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, 0, 3, -18, 2, 0, 11, 0, 0, -10, -5, 9, 7, -14, 0, 0, 0, 0, 11, 9
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OFFSET
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0,5
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COMMENTS
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LINKS
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FORMULA
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Expansion of q^(-5/8) * eta(q) * eta(q^4)^4 / eta(q^2) in powers of q.
Euler transform of period 4 sequence [-1, 0, -1, -4, ...].
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)).
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.
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EXAMPLE
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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 + ...
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 + ...
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MATHEMATICA
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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 *)
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PROG
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(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))};
(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)))))};
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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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