Seven weeks after the coup that led to the overthrow and executions of leaders of the Liberian government, the United States is sending a high-level mission to Monrovia this weekend to explore improved relations with the new rulers.

The mission, composed of Assistant Secretary of State Richard M. Moose, Rep. William H. Gray III (D-Pa.) and Jerry Funk of the National Security Council, is to arrive in Liberia Sunday for a stay of several days. State Department officials portrayed the mission as a symbol of Washington's willingness to improve relations with a West African nation where the United States has had a long and close relationship.

The April 12 coup by Sgt. Samuel K. Doe and several other noncommissioned army officers, and expecially the grisly execution of President William Tolbert and 13 of his cabinet, soured Washington-Monrovia relations and cast future ties in doubt.

The United States halted a Pentagon training mission which had been scheduled to arrive in Liberia on May 1, and facilitated the departure from the country of American dependents who were jittery about the situation. Moreover, questions were raised about the future of the $6 million economic aid program and the $1.4 million military credit sales program.

Announcement by the new regime on April 29 that there would be no further executions was met with relief here and in many other capitals which had called publicly or through diplomatic channels for an end to the killing.

An unannounced White House meeting of senior members of the National Security Council May 7 decided to send the Pentagon training team -- a matter of intense interest to the new Liberian leadership -- and move in other ways to patch up bilateral ties. The current mission to Monrovia flows out of the May 7 decision-making.

Before the coup, the United States had promised to send a goodwill commission named by President Carter to Liberia. The commission's trip was called off after the coup but Rep. Gray, previously designeated as commission chairman, expects to explore with the new Liberian leaders whether it should be reinstated.

State Department officials said it is yet unclear whether the new regime is stable enough or responsible enough to establish a close long-term relationship with Washington. These are among the questions which the Moose mission is to explore in Monrovia.

One reason for exploring closer ties, in the view of officials here, is a widespread belief in Africa that Washington bears special responsibility for anything that happens in Liberia.

Another reason is to forestall bids by unfriendly nations to become Liberia's foreign connection. Libyan leader Muammar Qaddafi sent an emissary to Monrovia this month with feelers for the Doe regime. The Soviet-backed Ethiopian regime also sent an emissary and last week received a return visit from Liberia's new foreign minister.