A292042 - OEIS

%I #37 Jan 19 2021 21:53:28

%S 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 37,45,48,57,60,68,70,77,76,82,78,80,72,70,55,48,26,11,-19,-42,-84,

%U -116,-169,-213,-278,-333,-413,-479,-572,-651,-757,-846,-965,-1062

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

%H Seiichi Manyama, <a href="/A292042/b292042.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>.

%F ( i*x; x)_inf is the g.f. for a(n) + i*A292043(n).

%F (-i*x; x)_inf is the g.f. for a(n) + i*A292052(n).

%F a(n)^2 + A292043(n)^2 = A278420(n). - _Vaclav Kotesovec_, Sep 08 2017

%F From _Peter Bala_, Jan 15 2021: (Start)

%F 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 Conjectural g.f.: A(x) = (1/2)*Sum_{n >= 0} (-x)^(n*(n-1)/2)/Product_{k = 1..n} (1 - x^k). (End)

%e 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 N:= 100:

%p 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 seq(coeff(S, x, n), n = 0..N); # _Peter Bala_, Jan 15 2021

%t Re[CoefficientList[Series[QPochhammer[I*x, x], {x, 0, 100}], x]] (* _Vaclav Kotesovec_, Sep 08 2017 *)

%Y Cf. A035294, A278399, A278400, A278420, A292043, A292052.

%K sign

%O 0,6

%A _Seiichi Manyama_, Sep 08 2017