A072338 EULER transform of A002620 (with the initial 0,0,1 omitted).

1, 2, 7, 18, 48, 114, 273, 614, 1370, 2952, 6275, 13034, 26725, 53854, 107238, 210670, 409446, 786936, 1498147, 2825084, 5282409, 9795620, 18027645, 32935112, 59760481, 107724038, 192984835, 343676216, 608589028, 1071869694, 1878068324, 3274291480, 5681336242