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A292042 G.f.: Re((i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

%I A292042 #38 Feb 16 2025 08:33:51
%S A292042 1,0,0,-1,-1,-2,-2,-3,-3,-4,-3,-4,-3,-3,-1,-1,2,3,7,9,14,16,23,26,33,
%T A292042 37,45,48,57,60,68,70,77,76,82,78,80,72,70,55,48,26,11,-19,-42,-84,
%U A292042 -116,-169,-213,-278,-333,-413,-479,-572,-651,-757,-846,-965,-1062
%N A292042 G.f.: Re((i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).
%H A292042 Seiichi Manyama, <a href="/A292042/b292042.txt">Table of n, a(n) for n = 0..1000</a>
%H A292042 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>.
%F A292042 ( i*x; x)_inf is the g.f. for a(n) + i*A292043(n).
%F A292042 (-i*x; x)_inf is the g.f. for a(n) + i*A292052(n).
%F A292042 a(n)^2 + A292043(n)^2 = A278420(n). - _Vaclav Kotesovec_, Sep 08 2017
%F A292042 From _Peter Bala_, Jan 15 2021: (Start)
%F A292042 G.f.: A(x) = Sum_{n >= 0} (-1)^n*x^(n*(2*n+1))/Product_{k = 1..2*n}  (1 - x^k). Cf. A035294.
%F A292042 Conjectural g.f.: A(x) = (1/2)*Sum_{n >= 0} (-x)^(n*(n-1)/2)/Product_{k = 1..n} (1 - x^k). (End)
%e A292042 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 A292042 N:= 100:
%p A292042 S := convert(series( add( (-1)^n*x^(n*(2*n+1))/(mul(1 - x^k,k = 1..2*n)), n = 0..floor(sqrt(N/2)) ), x, N+1 ), polynom):
%p A292042 seq(coeff(S, x, n), n = 0..N); # _Peter Bala_, Jan 15 2021
%t A292042 Re[CoefficientList[Series[QPochhammer[I*x, x], {x, 0, 100}], x]] (* _Vaclav Kotesovec_, Sep 08 2017 *)
%Y A292042 Cf. A035294, A278399, A278400, A278420, A292043, A292052.
%K A292042 sign
%O A292042 0,6
%A A292042 _Seiichi Manyama_, Sep 08 2017