Z:=SumTools[Hypergeometric][Zeilberger]: print(`A 0.14-Seconds proof of the Main Result of a 17-page Article that Appeared in the May 2017 volume of the journal Advances in Applied Mathematics`): print(``): print(`By Shalosh B. Ekhad `): print(``): print(`In this note we give one-line, automatic proofs of the main result (Theorem 1.3) of the paper: `): print(` "Identities involving weighted Catalan, Schroder, and Motkzin paths"`): print(`by Zhi Chen and Hao Pan, arXiv:1608.02448v2 [math.CO], that appeared recently in Advances in Applied Mathematics`): print(` v. 86 (May 2017), pp.81-98 `): ope1L:=Z(1/n*binomial(n,k)*binomial(n,k+1)*a^(n-k)*b^k,n,k,N)[1]: ope1R:=Z(a*binomial(2*k,k)/(k+1)*binomial(n-1,2*k)*(a+b)^(n-1-2*k)*(a*b)^k,n,k,N)[1]: print(`The left side of Eq. (1.14), let's call it L1(n), satisfies the recurrence`): print(``): print(add(factor(coeff(ope1L,N,i))*L1(n+i),i=0..degree(ope1L,N))=0): print(``): print(`The right side of Eq. (1.14), let's call it R1(n), satisfies the recurrence`): print(``): print(add(factor(coeff(ope1R,N,i))*R1(n+i),i=0..degree(ope1R,N))=0): print(``): print(`Since L1(0)=R1(0), and L1(1)=R1(1) (check!), and the two sequences satisfy the same second-order recurrence this proves (1.14) of the above paper`): print(``): print(`------------------------------------------------------------------`): print(``): ope2L:=Z(b*binomial(n+k,2*k)*binomial(2*k,k)/(k+1)*a^(n-k)*b^k,n,k,N)[1]: ope2R:=Z((a+b)*1/n*binomial(n,k)*binomial(n,k+1)*b^(n-k)*(a+b)^k,n,k,N)[1]: print(`The left side of Eq. (1.15), let's call it L2(n), satisfies the recurrence`): print(``): print(add(factor(coeff(ope2L,N,i))*L2(n+i),i=0..degree(ope2L,N))=0): print(``): print(`The right side of Eq. (1.15), let's call it R2(n), satisfies the recurrence`): print(``): print(add(factor(coeff(ope2R,N,i))*R2(n+i),i=0..degree(ope2R,N))=0): print(``): print(`Since L2(0)=R2(0), and L2(1)=R2(1) (check!), and the two sequences satisfy the same second-order recurrence this proves (1.15) of the above paper`): print(``): print(`------------------------------------------------------------------`): print(``): ope3L:=Z(binomial(n+k,2*k)*binomial(2*k,k)/(k+1)*a^(n-k)*b^k,n,k,N)[1]: ope3R:=Z((a+b)*binomial(2*k,k)/(k+1)*binomial(n-1,2*k)*(a+2*b)^(n-1-2*k)*(a*b+b^2)^k,n,k,N)[1]: print(`The left side of Eq. (1.16), let's call it L3(n), satisfies the recurrence`): print(``): print(add(factor(coeff(ope3L,N,i))*L3(n+i),i=0..degree(ope3L,N))=0): print(``): print(`The right side of Eq. (1.16), let's call it R3(n), satisfies the recurrence`): print(``): print(add(factor(coeff(ope3R,N,i))*R3(n+i),i=0..degree(ope3R,N))=0): print(``): print(`Since L3(0)=R3(0), and L3(1)=R3(1) (check!), and the two sequences satisfy the same second-order recurrence this proves (1.16) of the above paper`): print(``): print(`------------------------------------------------------------------`): print(``): print(`The whole thing took`, time(), `seconds. `): quit: