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A255252 Expansion of psi(x) * psi(-x)^2 in powers of x where psi() is a Ramanujan theta function.

%I A255252 #12 Feb 16 2025 08:33:25
%S A255252 1,-1,-1,0,-2,3,2,1,-1,-1,1,-2,1,-3,-2,-2,3,1,-1,4,3,-1,-1,2,-4,4,1,0,
%T A255252 -1,-2,-3,-3,-4,2,3,-3,0,0,5,2,0,-3,2,-1,4,1,0,1,3,0,-2,2,-1,-2,-4,-5,
%U A255252 2,0,-7,3,-4,3,1,5,2,-5,-1,-1,-3,4,-1,3,4,1,4
%N A255252 Expansion of psi(x) * psi(-x)^2 in powers of x where psi() is a Ramanujan theta function.
%C A255252 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A255252 G. C. Greubel, <a href="/A255252/b255252.txt">Table of n, a(n) for n = 0..1000</a>
%H A255252 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A255252 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A255252 Expansion of f(-x) * f(-x^4)^2 in powers of x where f() is a Ramanujan theta function.
%F A255252 Expansion of q^(-3/8) * eta(q) * eta(q^4)^2 in powers of q.
%F A255252 Euler transform of period 4 sequence [ -1, -1, -1, -3, ...].
%F A255252 G.f.: Product_{k>0} (1 - x^k) * (1 - x^(4*k))^2.
%F A255252 2 * a(n) = A034950(4*n + 1).
%e A255252 G.f. = 1 - x - x^2 - 2*x^4 + 3*x^5 + 2*x^6 + x^7 - x^8 - x^9 + x^10 + ...
%e A255252 G.f. = q^3 - q^11 - q^19 - 2*q^35 + 3*q^43 + 2*q^51 + q^59 - q^67 - q^75 + ...
%p A255252 A255252 := proc(n)
%p A255252     local psi,x,i ;
%p A255252     psi := add( A010054(i)*x^i,i=0..n) ;
%p A255252     psi*subs(x=-x,psi)^2 ;
%p A255252     coeftayl(%,x=0,n) ;
%p A255252 end proc:
%p A255252 seq(A255252(n),n=0..20) ; # _R. J. Mathar_, Feb 22 2021
%t A255252 a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^4]^2, {x, 0, n}];
%t A255252 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, Pi/4, x^(1/2)]^2 / (4 x^(3/8)), {x, 0, n}];
%o A255252 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^2, n))};
%Y A255252 Cf. A034950.
%K A255252 sign
%O A255252 0,5
%A A255252 _Michael Somos_, Feb 18 2015