A036719 G.f. satisfies A(x) = 1 + x*cycle_index(Cyclic(4), A(x)).

1, 1, 1, 3, 7, 22, 65, 212, 697, 2372, 8179, 28703, 101851, 365393, 1322034, 4820378, 17691018, 65306700, 242317423, 903242615, 3380707920, 12700575742, 47874113728, 181013422393, 686341240781, 2609093456212, 9941953711409