a question

[Emeric Deutsch]

Dear Seqfans,

The following is known:
Given a partition p of n into k parts, [a_1^e_1,...,a_j^e_j]
(e_1 + e_2 + ... + e_j = k), the number b(p) of Dyck paths of
semilength n whose ascent lengths yield the partition p is
    	b(p)=n!/[(n-k+1)!*e_1!*e_2!*...*e_j!].

Apparently, we have

      SUM(b(p)*SUM(e_i*binom(a_i + 1, 2), i=1..j) = binom(2n+1,n-1),

where the outer sum is over all partitions p of n.

For example, for n = 4, the partitions 4, 31, 22, 211, 1111 yield
      1*10 + 4*7 + 2*6 + 6*5 + 1*4 = 84.

Has anybody seen this? Any idea for a proof? Any information?
Many thanks.

Emeric