%I #12 Jul 29 2021 07:26:26
%S 0,1,-1,-1,2,2,-5,-5,14,14,-42,-42,132,132,-429,-429,1430,1430,-4862,
%T -4862,16796,16796,-58786,-58786,208012,208012,-742900,-742900,
%U 2674440,2674440,-9694845,-9694845,35357670,35357670,-129644790,-129644790,477638700,477638700,-1767263190,-1767263190
%N Expansion of c(-x^2)(1+2x-sqrt(1+4x^2))/2, c(x) the g.f. of A000108.
%C Second column of number triangle A106180.
%F a(n) = sin(Pi*n/2)*(C((n-1)/2)*(1-(-1)^n)/2) + sin(Pi*(n+1)/2)*(C(n/2)*(1+(-1)^n)/2) - 0^n for n > 0.
%F Conjecture: (n+2)*a(n) + n*a(n-1) + 4*(n-1)*a(n-2) + 4*(n-3)*a(n-3) = 0. - _R. J. Mathar_, Nov 15 2011
%Y Cf. A000108, A099363.
%K easy,sign
%O 0,5
%A _Paul Barry_, Apr 24 2005