In this paper we give a combinatorial proof of an addition formula for weighted partial Motzkin paths. The addition formula allows us to determine the $ LDU $ decomposition of a Hankel matrix of the polynomial sequence defined by weighted partial Motzkin paths. As a direct consequence, we get the determinant of the Hankel matrix of certain combinatorial sequences. In addition, we obtain an addition formula for weighted large Schrder paths.

This article studies the geometry of moduli spaces of G2-manifolds, associative cycles, coassociative cycles and deformed Donaldson–Thomas bundles. We introduce natural symmetric cubic tensors and differential forms on these moduli spaces. They correspond to Yukawa couplings and correlation functions in M-theory. We expect that the Yukawa coupling characterizes (co-)associative fibrations on these manifolds. We discuss the Fourier transformation along such fibrations and the analog of the Strominger–Yau–Zaslow mirror conjecture for G2-manifolds. We also discuss similar structures and transformations for Spin(7)- manifolds.

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