%I #12 Aug 19 2018 06:56:21
%S 3,3,1,3,59,5,43,7,139,9,-127,11,62099,13,-2513,15,278923,17,
%T -16761307,19,13372769,21,-327211439,23,18102814403,25,-655076773,27,
%U 1818961119031,29,-659884043593273,31,6494995052521753,33,-197424183925133,35,2015477183184289687757
%N a(n) = numerator(n!*[z^n](z*(exp(z)+2*exp(-(1/2)*z)*cos((1/2)*z* 3^(1/2)))/(1-exp(-z)))).
%F a(2*n - 3) = 2*n - 3 for n >= 3.
%e Rational values start: 3, 3/2, 1/2, 3, 59/10, 5, 43/14, 7, 139/10, 9, -127/22, 11, 62099/910, 13, -2513/10, 15, 278923/170, 17, ....
%p gf := z*(exp(z)+2*exp(-(1/2)*z)*cos((1/2)*z*sqrt(3)))/(1-exp(-z));
%p ser := series(gf, z, 100): seq(numer(n!*coeff(ser, z, n)),n=0..36);
%Y Cf. A306149 (denominator).
%K sign,frac
%O 0,1
%A _Peter Luschny_, Aug 19 2018