Jamieson, Fausset and Brown on 2 Samuel 12:15-23

15-23. the Lord struck the child ... and it was very sick--The first visible chastisement inflicted on David appeared on the person of that child which was the evidence and monument of his guilt. His domestics were surprised at his conduct, and in explanation of its singularity, it is necessary to remark that the custom in the East is to leave the nearest relative of a deceased person to the full and undisturbed indulgence of his grief, till on the third or fourth day at farthest (Joh 11:17). Then the other relatives and friends visit him, invite him to eat, lead him to a bath, and bring him a change of dress, which is necessary from his having sat or lain on the ground. The surprise of David's servants, then, who had seen his bitter anguish while the child was sick, arose apparently from this, that when he found it was dead, he who had so deeply lamented arose of himself from the earth, without waiting for their coming to him, immediately bathed and anointed himself, instead of appearing as a mourner, and after worshiping God with solemnity, returned to his wonted repast, without any interposition of others.

2Sa 12:24, 25. Solomon Is Born.