In this paper,by means of the classical Lagrange inversion formula,the authors establish a general nonlinear inverse relation as the solution to the problem proposed in the paper[J.Wang,Nonlinear inverse relations for...In this paper,by means of the classical Lagrange inversion formula,the authors establish a general nonlinear inverse relation as the solution to the problem proposed in the paper[J.Wang,Nonlinear inverse relations for the Bell polynomials via the Lagrange inversion formula,J.Integer Seq.,Vol.22(2019),Article 19.3.8].As applications of this inverse relation,the authors not only find a short proof of another nonlinear inverse relation due to Birmajer,et al.(2012),but also set up a few convolution identities concerning the Mina polynomials.展开更多
The present paper gives a new Bell matrix inversion which arises from the classical Lagrange inversion formula. Some new relations for the Bell polynomials are obtained, including a Bell matrix inversion in closed for...The present paper gives a new Bell matrix inversion which arises from the classical Lagrange inversion formula. Some new relations for the Bell polynomials are obtained, including a Bell matrix inversion in closed form and an inverse form of the classical Faa di Bruno formula.展开更多
In this paper,we enumerate the set of Motzkin trees with n edges according to the number of leaves,the number of vertices adjacent to a leaf,the number of protected nodes,the number of(protected)branch nodes,and the n...In this paper,we enumerate the set of Motzkin trees with n edges according to the number of leaves,the number of vertices adjacent to a leaf,the number of protected nodes,the number of(protected)branch nodes,and the number of(protected)lonely nodes.Explicit formulae as well as generating functions are obtained.We also find that,as n goes to infinity,the proportion of protected branch nodes and protected lonely nodes among all vertices of Motzkin trees with n edges approaches 4/27 and 2/9.展开更多
We consider some new alternating double binomial sums. By using the Lagrange inversion formula, we obtain explicit expressions of the desired results which are related to a third-order linear recursive sequence. Furth...We consider some new alternating double binomial sums. By using the Lagrange inversion formula, we obtain explicit expressions of the desired results which are related to a third-order linear recursive sequence. Furthermore, their recursive relation and generating functions are obtained.展开更多
基金supported by the National Natural Science Foundation of China under Grant Nos.11971341 and 12001492the Natural Science Foundation of Zhejiang Province under Grant No.LQ20A010004.
文摘In this paper,by means of the classical Lagrange inversion formula,the authors establish a general nonlinear inverse relation as the solution to the problem proposed in the paper[J.Wang,Nonlinear inverse relations for the Bell polynomials via the Lagrange inversion formula,J.Integer Seq.,Vol.22(2019),Article 19.3.8].As applications of this inverse relation,the authors not only find a short proof of another nonlinear inverse relation due to Birmajer,et al.(2012),but also set up a few convolution identities concerning the Mina polynomials.
基金Supported by the National Natural Science Foundation of China(Grant No.11401237)
文摘The present paper gives a new Bell matrix inversion which arises from the classical Lagrange inversion formula. Some new relations for the Bell polynomials are obtained, including a Bell matrix inversion in closed form and an inverse form of the classical Faa di Bruno formula.
基金Supported by the National Natural Science Foundation of China(Grant No.11861045)Gansu Province Science Foundation for Youths(Grant No.20JR10RA187)the Hongliu Foundation of First-Class Disciplines of Lanzhou University of Technology,China。
文摘In this paper,we enumerate the set of Motzkin trees with n edges according to the number of leaves,the number of vertices adjacent to a leaf,the number of protected nodes,the number of(protected)branch nodes,and the number of(protected)lonely nodes.Explicit formulae as well as generating functions are obtained.We also find that,as n goes to infinity,the proportion of protected branch nodes and protected lonely nodes among all vertices of Motzkin trees with n edges approaches 4/27 and 2/9.
基金Foundation item： Supported by the Natural Science Foundation of Zhejiang Province（YT080320, LYI2A01030） Supported by the National Natural Science Foundation of China（l1226297） Supported by the Zhejiang Univerity City College Scientific Research Foundation（J-13003）
文摘We consider some new alternating double binomial sums. By using the Lagrange inversion formula, we obtain explicit expressions of the desired results which are related to a third-order linear recursive sequence. Furthermore, their recursive relation and generating functions are obtained.
