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A226075 Expansion of (eta(q) * eta(q^11))^2 + 2 * (eta(q^2) * eta(q^22))^2 in powers of q.

%I A226075 #19 Sep 08 2022 08:46:05
%S A226075 1,0,-1,-2,1,0,-2,4,-2,0,1,2,4,0,-1,-4,-2,0,0,-2,2,0,-1,-4,-4,0,5,4,0,
%T A226075 0,7,0,-1,0,-2,4,3,0,-4,4,-8,0,-6,-2,-2,0,8,4,-3,0,2,-8,-6,0,1,-8,0,0,
%U A226075 5,2,12,0,4,8,4,0,-7,4,1,0,-3,-8,4,0,4,0,-2,0
%N A226075 Expansion of (eta(q) * eta(q^11))^2 + 2 * (eta(q^2) * eta(q^22))^2 in powers of q.
%H A226075 G. C. Greubel, <a href="/A226075/b226075.txt">Table of n, a(n) for n = 1..2500</a>
%F A226075 a(n) is multiplicative with a(11^e) = 1, a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) if p != 11.
%F A226075 G.f. is a period 1 Fourier series which satisfies f(-1 / (22 t)) = 22 (t/i)^2 f(t) where q = exp(2 Pi i t).
%F A226075 a(4*n + 2) = 0. a(4*n) = -2 * A006571(n). a(2^n) = A090132(n). a(3^n) = A214733(n+1).
%e A226075 G.f. = q - q^3 - 2*q^4 + q^5 - 2*q^7 + 4*q^8 - 2*q^9 + q^11 + 2*q^12 + 4*q^13 + ...
%t A226075 a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^11])^2 + 2 q^2 ( QPochhammer[ q^2] QPochhammer[ q^22])^2, {q, 0, n}]; (* _Michael Somos_, Apr 25 2015 *)
%o A226075 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^11 + A))^2 + 2 * x * (eta(x^2 + A) * eta(x^22 + A))^2, n))};
%o A226075 (Sage) CuspForms( Gamma0(22), 2, prec=79).0;
%o A226075 (Magma) Basis( CuspForms( Gamma0(22), 2), 79)[1];
%Y A226075 Cf. A006571, A090132, A214733.
%K A226075 sign,mult
%O A226075 1,4
%A A226075 _Michael Somos_, May 25 2013