cp's OEIS Frontend

A226192 Expansion of phi(x^2) * psi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.

%I A226192 #12 Feb 16 2025 08:33:19
%S A226192 1,-1,2,-3,0,-2,1,0,4,-2,1,-2,2,0,2,-1,0,-2,4,-2,0,-3,0,-4,2,0,0,0,3,
%T A226192 -2,2,0,2,-4,0,-2,3,0,4,-2,0,0,2,0,2,-1,2,-4,0,0,2,-2,0,-6,2,-1,2,-2,
%U A226192 0,0,4,0,0,-4,0,-2,1,0,4,0,0,-2,2,-4,2,-2,0,-2
%N A226192 Expansion of phi(x^2) * psi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.
%C A226192 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A226192 G. C. Greubel, <a href="/A226192/b226192.txt">Table of n, a(n) for n = 0..1000</a>
%H A226192 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A226192 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A226192 Expansion of f(x^2)^2 * chi(-x) in powers of x where chi(), f() are Ramanujan theta functions.
%F A226192 Expansion of q^(-1/8) * eta(q) * eta(q^4)^6 / (eta(q^2)^3 * eta(q^8)^2) in powers of q.
%F A226192 Euler transform of period 8 sequence [ -1, 2, -1, -4, -1, 2, -1, -2, ...].
%F A226192 a(n) = b(8*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (-1)^(e * (p+1)/8) * (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = (e+1) * I^(e * (p-1)/4) if p == 1 (mod 4).
%F A226192 a(9*n + 4) = a(9*n + 7) = 0. a(9*n + 1) = -a(n).
%F A226192 a(n) = (-1)^n * A113407(n).
%e A226192 1 - x + 2*x^2 - 3*x^3 - 2*x^5 + x^6 + 4*x^8 - 2*x^9 + x^10 - 2*x^11 + ...
%e A226192 q - q^9 + 2*q^17 - 3*q^25 - 2*q^41 + q^49 + 4*q^65 - 2*q^73 + q^81 - 2*q^89 + ...
%t A226192 a[n_]:= SeriesCoefficient[QPochhammer[-q^2, -q^2]^2*QPochhammer[q, q^2], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Dec 07 2017 *)
%o A226192 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^6 / (eta(x^2 + A)^3 * eta(x^8 + A)^2), n))}
%o A226192 (PARI) {a(n) = local(A, p, e); if( n<0, 0, n = 8*n + 1; A = factor(n); simplify( prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, if( p%4 == 3, if( e%2, 0, (-1)^(e * (p+1) / 8)), (e+1) * I^(e * (p-1)/ 4)))))))}
%Y A226192 Cf. A113407, A226194.
%K A226192 sign
%O A226192 0,3
%A A226192 _Michael Somos_, May 30 2013