A Proof of the above Briggs Identity using the Lagrange Inversion Theorem

David M. Jackson ( Dept. of Combinatorics and Optimization, University of Waterloo, Ontario.), has been kind enough to provide an independent proof of the above Briggs-related identity. His proof makes use of a form of the Lagrange Inversion Theorem, (see e.g. E.T. Whittaker and G. N. Watson, A Course on Modern Analysis, 4th edition, C.U.P., 1950, p. 130.), adapted for the ring of formal power series.
First some notation: The ring of all formal power series in an indeterminate over the rationals is denoted by Q[[]]. The elements of this ring are infinite series that do not necessarily converge. If f Q[[]], then [ⁿ] f denotes the coefficient of ⁿ in f. The statement of the theorem for Lagrange inversion of Q[[]] now follows:

Let () be a formal power series in such that () 0. Let F() be another formal power series in . Then the functional equation w = t(w) has a unique solution. Moreover, if n is a non-negative integer, then

F(w) = F(0) + (tⁿ/n)[^n-1]F'()ⁿ(.

An example:
Consider the functional equation T = x e^T for T(x). By the theorem there is a unique series T(x) Q[[x]] that satisfies this equation, and [xⁿ]T =(1/n)[^n-1](^d/_d)eⁿ = n^n-1/n!, hence T(x) = (n^n-1/n!)xⁿ, where F(w) = w in this simple case.
Again, if we have T⁵ = x e^T, then [xⁿ] = (1/n)[^n-1](^d/_d⁵)eⁿ = (5/n)n^n-5/(n-5)!, and T⁵(x) = 5 (1/n)(n^n-5/(n-5)!)xⁿ.

Now to the Briggs-related identity:
Let r be a non-negative integer, then _N-iC_{r-i 2N+1}C_2i-1 = 2^2r(_N+r+1C_N-r + _N+rC_N-r-1.
Proof
Let _r,N denote the summation on the left hand side of the statement. Then, on two applications of the binomial theorem,

_r,N = ([t^r-1](1 + t)^N-i)([x²ⁱ⁺¹](1 + x)^2N+1) = [t^r](1 + t)^N tⁱ/(1+t)ⁱ([x²ⁱ](1 + x)^2N+1)/x =
[t^r](1 + t)^N((1+ (t/(1+t)))^2N+1)/(t/(1+t))

_r,N = [t^2r+1](t + (1+t²))^2N+1

Hence, in the Lagrange Inversion Theorem, we set w = 2t(1+w) = 2t(w) where (w) = 2(1 + w), and the required expansion F(w) = (1 + w)^N+1/2, to give

_r,N = [t^2r+1](q)^2N+1 = [w^2r+1](1 + w)^N+1/2 = (1/(2r+1))[^2r](N+1/2)(1 + )^N-1/2 2^2r+1((1+))^2r+1 =
2^2r(2N + 1)/(2r + 1) [^2r](1 + )^N+r = 2^2r(2N + 1)/(2r + 1) _N+rC_2r = 2^2r(_N+rC_2r+1 + _N+r+1C_2r+1)

If you are interested in this sort of thing, and would like to become more conversant with these remarkable short-hand methods, then Combinatorial Enumeration, by I. P. Goulden and D. M. Jackson, Wiley (1983) is a text well worth examining. Lagrange's Theorem in the single variable form is on page 17, with further generalisations following and other examples.