A096592 Row sums of triangle A096591, which shifts one place diagonally left and upward under the matrix square operation.

1, 2, 3, 5, 8, 15, 28, 61, 133, 328, 807, 2173, 5847, 16794, 48331, 145815, 441838, 1385737, 4371171, 14143062, 46053372, 152839735, 510525852, 1730174514, 5900106553, 20347687842, 70580227781, 247012709141, 869062063044