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A292043 G.f.: Im((i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

%I A292043 #37 Feb 16 2025 08:33:51
%S A292043 0,-1,-1,-1,-1,-1,0,0,1,2,3,4,6,7,9,10,12,13,15,15,16,16,16,14,13,9,6,
%T A292043 0,-5,-14,-22,-34,-45,-60,-74,-93,-110,-132,-152,-177,-199,-226,-249,
%U A292043 -277,-300,-328,-348,-373,-389,-408,-417,-428,-425,-424,-407,-389,-352
%N A292043 G.f.: Im((i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).
%H A292043 Seiichi Manyama, <a href="/A292043/b292043.txt">Table of n, a(n) for n = 0..1000</a>
%H A292043 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>.
%F A292043 (i*x; x)_inf is the g.f. for A292042(n) + i*a(n).
%F A292043 A292042(n)^2 + a(n)^2 = A278420(n). - _Vaclav Kotesovec_, Sep 08 2017
%F A292043 From _Peter Bala_, Feb 05 2021: (Start)
%F A292043 G.f.: Sum_{n >= 0} (-1)^(n+1)*x^((n+1)*(2*n+1))/Product_{k = 1..2*n+1} (1 - x^k).
%F A292043 The 2 X 2 matrix Product_{k >= 1} [1, -x^k; x^k, 1] = [A(x), B(x); -B(x), A(x)], where A(x) is the g.f. of A292042 and B(x) is the g.f. for this sequence.
%F A292043 A(x)^2 + B(x)^2 = Product_{k >= 1} 1 + x^(2*k) = A000009(x^2).
%F A292043 A(x) + B(x) is the g.f. of A278399; B(x) - A(x) is the g.f. of A278400. (End)
%e A292043 Product_{k>=1} (1 - i*x^k) = 1 + (0-1i)*x + (0-1i)*x^2 + (-1-1i)*x^3 + (-1-1i)*x^4 + (-2-1i)*x^5 + (-2+0i)*x^6 + (-3+0i)*x^7 + ...
%p A292043 N:= 100:
%p A292043 S := convert(series( add( (-1)^(n+1)*x^((n+1)*(2*n+1))/(mul(1 - x^k,k = 1..2*n+1)), n = 0..floor(sqrt(N/2)) ), x, N+1 ), polynom):
%p A292043 seq(coeff(S, x, n), n = 0..N); # _Peter Bala_, Feb 05 2021
%t A292043 Im[CoefficientList[Series[QPochhammer[I*x, x], {x, 0, 100}], x]] (* _Vaclav Kotesovec_, Sep 08 2017 *)
%Y A292043 Cf. A278399, A278400, A278420, A292042, A292052.
%K A292043 sign
%O A292043 0,10
%A A292043 _Seiichi Manyama_, Sep 08 2017