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In this paper, a second-order nonlinear recursive sequence <inline-formula> <math display="inline"> <semantics> <mrow> <mi>M</mi> <mo>(</mo> <mi>h</mi> <mo>,</mo> <mi>i</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is studied. By using this sequence, the properties of the power series, and the combinatorial methods, some interesting symmetry identities of the structural properties of balancing numbers and balancing polynomials are deduced.