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A254612 Expansion of f(-x) * f(-x^9) in powers of x where f() is a Ramanujan theta function.

%I A254612 #16 Feb 16 2025 08:33:24
%S A254612 1,-1,-1,0,0,1,0,1,0,-1,1,1,-1,0,-1,-1,-1,0,-1,1,1,1,1,-1,1,-1,1,0,0,
%T A254612 0,1,-1,0,1,0,-2,0,0,0,0,-2,0,0,0,0,1,-1,-1,0,1,1,1,1,1,0,0,0,0,1,0,
%U A254612 -2,0,0,1,-1,-1,-1,1,1,-1,0,1,0,0,0,-2,0,-1,-1
%N A254612 Expansion of f(-x) * f(-x^9) in powers of x where f() is a Ramanujan theta function.
%C A254612 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A254612 G. C. Greubel, <a href="/A254612/b254612.txt">Table of n, a(n) for n = 0..2500</a>
%H A254612 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A254612 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A254612 Expansion of q^(-5/12) * eta(q) * eta(q^9) in powers of q.
%F A254612 Euler transform of period 9 sequence [-1, -1, -1, -1, -1, -1, -1, -1, -2, ...].
%F A254612 G.f. is a period 1 Fourier series which satisfies f(-1 / (1296 t)) = 36 (t/i) f(t) where q = exp(2 Pi i t).
%F A254612 a(n) = b(12*n + 5) / sqrt(3) and A254613(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 if p == 7,11 (mod 12), b(p^e) = b(p)*b(p^(e-1)) - b(p^(e-2)) if p == 1,5 (mod 12) where b(p) = sqrt(3) * k12(x, y) * (if 3|y then 0 else 1) with 2*p = x^2 + 9*y^2 if p == 5 (mod 12) and b(p) = k12(9*y+x, y-x) + k12(9*y-x, y+x) with p = x^2 + 9*y^2 if p == 1 (mod 12) where k12(x,y) := Kronecker(12, x) * Kronecker(12, y). - _Michael Somos_, Feb 04 2015
%F A254612 G.f.: Product_{k>0} (1 - x^k) * (1 - x^(9*k)).
%F A254612 a(49*n + 20) = a(121*n + 50) = a(n).
%F A254612 A254613(n) = a(5*n) + (if n mod 5 = 2 then a((n-2)/5) otherwise 0). - _Michael Somos_, Feb 04 2015
%e A254612 G.f. = 1 - x - x^2 + x^5 + x^7 - x^9 + x^10 + x^11 - x^12 - x^14 - x^15 + ...
%e A254612 G.f. = q^5 - q^17 - q^29 + q^65 + q^89 - q^113 + q^125 + q^137 - q^149 + ...
%t A254612 a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^9], {x, 0, n}];
%o A254612 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^9 + A), n))};
%o A254612 (PARI) {a(n) = my(x); if( n<0, 0, n = 12*n + 5; sum(y=1, sqrtint( 2*n\9), if( issquare( 2*n - 9*y^2, &x), kronecker( 12, x) * kronecker( 12, y))))};
%o A254612 (PARI) {a(n) = my(A, p, e, ap, w, k12, x, y, xy); if( n<0, 0, n = 12*n + 5; w = quadgen(12); (k12 = (u,v) -> kronecker( 12,u)*kronecker( 12,v)); (xy = (m) -> if(1, my(x); for(i=1, sqrtint( m\9), if( issquare( m - 9*i^2, &x), return([x,i]))))); A = factor(n); 1/w * prod(k=1, matsize(A)[1], p = A[k,1]; e = A[k,2]; if( p<5, 0, p%12==7 || p%12==11, !(e%2), ap = if( p%12==5, [x,y] = xy(2*p); if(y%3==0, 0, w*k12(x,y)), [x,y] = xy(p); k12(9*y + x, y-x) + k12(9*y - x, y+x)); polchebyshev(e, 2, ap/2))))}; /* _Michael Somos_, Feb 04 2015 */
%Y A254612 Cf. A254613.
%K A254612 sign
%O A254612 0,36
%A A254612 _Michael Somos_, Feb 03 2015