number triangle related to row sums of pascal's triangle squared A038207

[Ross La Haye]

I'm looking at a number triangle a(n,k) that is constructed in the following
manner: column 0 is the row sums of A038207 starting at column 0, column 1
is the row sums of the same starting at column etc...so that the first three
columns are 3^n (A000244), 3^n - 2^n (A001047), 3^n - 2^n - n*2^(n-1)
(A066810).  In other words, a(n,k) = Sum[Binomial[n,k+m]*2^(n-k-m),{m,0,n}].
Computation and examination of the 3 sequences just listed strongly suggests
that the ogf by columns for this triangle would be x^k / (1-3x)(1-2x)^k, but
I'm having problems showing this.  Also, computation strongly suggests that
the sums of the rows of this triangle = n*3^(n-1) + 3^n A006234(n+3).  I'm
having problems showing this also.  Both problems seem like they should be
fairly easy, but I'm stuck...Can anyone help?

Ross


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