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%I #18 Feb 03 2023 14:36:33
%S 1,238,1679,2162,2401,-6958,-1442,-23040,1918,-9362,14641,0,80640,
%T -20398,28083,64078,-38398,-69120,0,-90482,-58562,0,-241920,100558,
%U 146879,0,-193438,399602,104638,114002,130321,24242,0,107282,-276962,351118
%N Expansion of f(-q)^2 * Q(q) in powers of q.
%C f(-q) (g.f. A010815) and Q(q) (g.f. A004009) are Ramanujan q-series.
%H S. R. Finch, <a href="http://arXiv.org/abs/math.NT/0701251">Powers of Euler's q-Series</a>, (arXiv:math.NT/0701251).
%F Expansion of q^(-1/12) * (eta(q)^16 + 256 * eta(q)^8 * eta(q^4)^8 + 4096 * eta(q^4)^16) * eta(q)^2 / eta(q^2)^8 in powers of q. - _Michael Somos_, Jun 25 2013
%F a(n) = b(12*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 7 or 11 (mod 12), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) if p == 1 or 5 (mod 12). - _Michael Somos_, Jun 24 2013
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 12^5 (t/i)^5 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Jan 06 2014
%F Convolution of A002107 and A004009. - _Michael Somos_, Jun 25 2013
%F Expansion of q^(-1/12) * (eta(q)^24 + 256*eta(q^2)^24) / (eta(q)^6*eta(q^2)^8) = q^(-1/12) * (eta(q)^12 + 250*eta(q)^6*eta(q^5)^6 + 3125*eta(q^5)^12) / eta(q^5)^2 in powers of q. - _Michael Somos_, Feb 03 2023
%e G.f. = 1 + 238*x + 1679*x^2 + 2162*x^3 + 2401*x^4 - 6958*x^5 - 1442*x^6 + ...
%e G.f. = q + 238*q^13 + 1679*q^25 + 2162*q^37 + 2401*q^49 - 6958*q^61 - 1442*q^73 + ...
%t a[ n_] := SeriesCoefficient[ (1 + 240 Sum[ DivisorSigma[ 3, k] q^k, {k, n}]) QPochhammer[ q]^2, {q, 0, n}]; (* _Michael Somos_, Jun 24 2013 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 (EllipticTheta[ 4, 0, q]^8 + EllipticTheta[ 2, 0, q^(1/2)]^8), {q, 0, n}]; (* _Michael Somos_, Jun 25 2013 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * sum( k=1, n, 240 * sigma( k, 3) * x^k, 1 + A), n))};
%o (PARI) {a(n) = my(A, B); if( n<0, 0, A = x * O(x^n); B = 64 * x * (eta(x^4 + A) / eta(x + A))^8; polcoeff( (1 + 4*B + B^2) * eta(x + A)^18 / eta(x^2 + A)^8, n))}; /* _Michael Somos_, Jun 25 2013 */
%o (PARI) {a(n) = my(A, p, e, i, x, y, a0, a1); if( n<0, 0, n = 12*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p<5, 0, if( p%12 > 6, if( e%2, 0, p^(2*e)), forstep( i = 1, sqrtint( p), 2, if( issquare( p - i^2, &y), x=i; break)); if( p%12 == 5, a1 = 8 * x*y * (x-y) * (x+y) * (-1)^((x%6==1) + (y%6==4)), a1 = 2 * (x^2-y^2+2*x*y) * (x^2-y^2-2*x*y) * (-1)^(x%6==3) ); a0 = 1; y = a1; for( i=2, e, x = y * a1 - p^4 * a0; a0=a1; a1=x); a1 )))))}; /* _Michael Somos_, Jun 24 2013 */
%Y Cf. A002107, A004009, A010818.
%K sign
%O 0,2
%A _Michael Somos_, Aug 28 2006
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