Total[Array[a, {3, 3}]*Array[b, {3, 3, 3}]]
(* {{a[1, 1] b[1, 1, 1] + a[2, 1] b[2, 1, 1] +
a[3, 1] b[3, 1, 1],
a[1, 1] b[1, 1, 2] + a[2, 1] b[2, 1, 2] + a[3, 1] b[3, 1, 2],
a[1, 1] b[1, 1, 3] + a[2, 1] b[2, 1, 3] +
a[3, 1] b[3, 1, 3]}, {a[1, 2] b[1, 2, 1] + a[2, 2] b[2, 2, 1] +
a[3, 2] b[3, 2, 1],
a[1, 2] b[1, 2, 2] + a[2, 2] b[2, 2, 2] + a[3, 2] b[3, 2, 2],
a[1, 2] b[1, 2, 3] + a[2, 2] b[2, 2, 3] +
a[3, 2] b[3, 2, 3]}, {a[1, 3] b[1, 3, 1] + a[2, 3] b[2, 3, 1] +
a[3, 3] b[3, 3, 1],