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A136599 Expansion of (eta(q) * eta(q^15))^3 in powers of q.

%I A136599 #5 Sep 08 2022 08:45:32
%S A136599 1,-3,0,5,0,0,-7,0,0,0,9,0,0,0,0,-14,9,0,-15,0,0,34,0,0,0,-27,0,0,-15,
%T A136599 0,33,0,0,0,0,0,-22,0,0,0,0,0,0,45,0,-14,-15,0,25,0,0,-86,0,0,0,66,0,
%U A136599 0,0,0,2,0,0,0,0,0,42,0,0,0,-63,0,0,-75,0,0,0,0
%N A136599 Expansion of (eta(q) * eta(q^15))^3 in powers of q.
%F A136599 Euler transform of period 15 sequence [ -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -6, ...].
%F A136599 G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = 15^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t).
%F A136599 a(n) nonzero or n=0 if and only if n is in A028955.
%F A136599 G.f.: x^2 * (Product_{k>0} (1 - x^k) * (1 - x^(15*k)))^3.
%F A136599 a(3*n) = -3 * A030220(n). a(3*n + 1) = 0. - _Michael Somos_, Oct 13 2015
%F A136599 A115155(n) = a(n) + A030220(n). - _Michael Somos_, Oct 13 2015
%e A136599 G.f. = q^2 - 3*q^3 + 5*q^5 - 7*q^8 + 9*q^12 - 14*q^17 + 9*q^18 - 15*q^20 + ...
%t A136599 a[ n_] := SeriesCoefficient[ (QPochhammer[ q] QPochhammer[ q^15])^3, {q, 0, n}]; (* _Michael Somos_, Oct 13 2015 *)
%o A136599 (PARI) {a(n) = my(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^15 + A))^3, n))};
%o A136599 (Magma) A := Basis( CuspForms( Gamma1(15), 3), 80); A[2] - 3*A[3] + 5*A[5] - 7*A[8]; /* _Michael Somos_, Oct 13 2015 */
%Y A136599 Cf. A028955, A030220, A115155.
%K A136599 sign
%O A136599 2,2
%A A136599 _Michael Somos_, Jan 11 2008