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A214960 Expansion of psi(x^2) - x * psi(x^10) in powers of x where psi() is a Ramanujan theta function.

%I A214960 #17 Aug 30 2025 21:30:33
%S A214960 1,-1,1,0,0,0,1,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,-1,0,
%T A214960 0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,
%U A214960 0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0
%N A214960 Expansion of psi(x^2) - x * psi(x^10) in powers of x where psi() is a Ramanujan theta function.
%C A214960 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A214960 G. C. Greubel, <a href="/A214960/b214960.txt">Table of n, a(n) for n = 0..1000</a>
%H A214960 R. Blecksmith; J. Brillhart; I. Gerst, <a href="http://dx.doi.org/10.1090/S0025-5718-1988-0942157-2">Some infinite product identities</a>, Math. Comp. 51 (1988), no. 183, 301-314. see p. 310.
%H A214960 S. Cooper and M. Hirschhorn, <a href="http://dx.doi.org/10.1216/rmjm/1008959672">On some infinite product identities</a>, Rocky Mountain J. Math., 31 (2001) 131-139. see p. 134 Theorem 6.
%H A214960 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A214960 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A214960 Expansion of f(-x, -x^4) * f(-x^2, x^3) / f(-x^2, -x^2) = f(-x, x^4) * f(x^2, x^3) / f(-x^10, -x^10) in powers of x where f(,) is Ramanujan's two-variable theta function.
%F A214960 Euler transform of period 20 sequence [ -1, 1, 1, 0, 0, 0, 1, 0, -1, -1, -1, 0, 1, 0, 0, 0, 1, 1, -1, -1, ...].
%F A214960 a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(5^e) = (-1)^e, else b(p^e) = (1 + (-1)^e) / 2.
%F A214960 G.f.: Sum_{k>0} x^(k*(k - 1)) - x^(5*k*(k - 1) + 1) = Product_{k>0} (1 - x^(10*k)) * (1 - x^(10*k - 1)) * (1 + x^(10*k-2)) * (1 + x^(10*k - 3)) * (1 + x^(10*k - 4)) * (1 + x^(10*k - 6)) * (1 + x^(10*k - 7)) * (1 + x^(10*k - 8)) * (1 - x^(10*k - 9)).
%F A214960 a(9*n + 2) = - a(5*n + 1) = a(n), a(5*n + 3) = a(5*n + 4) = a(6*n + 3) = a(6*n + 4) = a(9*n + 5) = a(9*n + 8) = 0.
%F A214960 a(n) = (-1)^n * A127693(n). a(2*n) = A010054(n). a(3*n) = A089806(n). a(6*n) = A080995(n).
%e A214960 G.f. = 1 - x + x^2 + x^6 - x^11 + x^12 + x^20 + x^30 - x^31 + x^42 + x^56 - x^61 + ...
%e A214960 G.f. = q - q^5 + q^9 + q^25 - q^45 + q^49 + q^81 + q^121 - q^125 + q^169 + q^225 + ...
%t A214960 f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A214960[n_] := SeriesCoefficient[f[-x, -x^4]*f[-x^2, x^3]/f[-x^2, -x^2], {x, 0, n}]; Table[A214960[n], {n,0,50}] (* _G. C. Greubel_, Jun 18 2017 *)
%o A214960 (PARI) {a(n) = issquare(4*n + 1) - issquare(20*n + 5)};
%o A214960 (PARI) {a(n) = local(A, p, e); if( n<0, 0, A = factor(4*n + 1); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, if( p==5, (-1)^e, (1 + (-1)^e) / 2)))))};
%Y A214960 Cf. A010054, A080995, A089806, A127693.
%K A214960 sign,changed
%O A214960 0,1
%A A214960 _Michael Somos_, Jul 30 2012