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A208845 Expansion of f(x)^2 in powers of x where f() is a Ramanujan theta function.

%I A208845 #24 Feb 16 2025 08:33:16
%S A208845 1,2,-1,-2,1,-2,-2,0,-2,2,1,0,0,-2,3,2,2,0,0,2,-2,0,0,2,-1,0,2,-2,-2,
%T A208845 -2,1,-2,0,-2,-2,2,2,0,-2,0,-4,0,0,0,1,2,0,0,2,0,2,-2,1,2,0,-2,2,0,0,
%U A208845 2,0,2,0,2,2,0,-4,0,0,2,-1,-2,0,-2,0,0,0,2,2
%N A208845 Expansion of f(x)^2 in powers of x where f() is a Ramanujan theta function.
%C A208845 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A208845 Number 73 of the 74 eta-quotients listed in Table I of Martin (1996).
%H A208845 Seiichi Manyama, <a href="/A208845/b208845.txt">Table of n, a(n) for n = 0..10000</a>
%H A208845 Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H A208845 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H A208845 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A208845 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A208845 Expansion of q^(-1/12) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^2 in powers of q.
%F A208845 Euler transform of period 4 sequence [ 2, -4, 2, -2, ...].
%F A208845 G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = (24)^(1/2) (t/i)^(1/2) f(t) where q = exp(2 Pi i t).
%F A208845 a(n) = b(12*n + 1) where b(n) is multiplicative with  b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12), b(p^e) = (-1)^(e/2) * (1 + (-1)^e) / 2 if p == 5 (mod 12), b(p^e) = (e + 1) * (-1)^(e * ((p%24>1) + x)) if p == 1 (mod 12) and p = x^2 + 9 * y^2.
%F A208845 a(n) = (-1)^n * A002107(n). a(25*n + 2) = -a(n).
%F A208845 Convolution cube is A209941. - _Michael Somos_, Jun 09 2015
%e A208845 G.f. = 1 + 2*x - x^2 - 2*x^3 + x^4 - 2*x^5 - 2*x^6 - 2*x^8 + 2*x^9 + x^10 + ...
%e A208845 G.f. = q + 2*q^13 - q^25 - 2*q^37 + q^49 - 2*q^61 - 2*q^73 - 2*q^97 + ...
%t A208845 a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^2 , {x, 0, n}]; (* _Michael Somos_, Jun 09 2015 *)
%o A208845 (PARI) {a(n) = my(A, p, e, m); if( n<0, 0, n = 12*n + 1; A=factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, 0, p%12>1, if( e%2, 0, (-1)^( (p%12==5) * e/2)), for( i=1, sqrtint(p\9), if( issquare( p - 9*i^2), m=i; break)); (e+1) * (-1)^(e * ( (p%24>1) + m )))))};
%o A208845 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A) * eta(x^4 + A)))^2, n))};
%Y A208845 Cf. A002107, A209941.
%K A208845 sign
%O A208845 0,2
%A A208845 _Michael Somos_, Mar 03 2012