A binomial sum

[Henry Gould]

Gentlemen -
If we tabulate ANY function, as e.g. a Table of Prime numbers, or admit a
Hypergeometric function as a tabulated function, then of course we have a
"closed form".
We can express the product of the even nunbers 2,4,6,...2n by the "closed
form" (2^n)n!.
We can express the product of the odd numbers 1,3,5,...2n-1 by the "closed
form" (2n)!/(2^n)n!,
but we cannot do the same with the product of 1,4,7,10,13,16,...,3n-2.  Of
course if we admit another this as a tabulated function and assign it a
name, e.g. P(n,3), then we ipso facto have a "formula."
It is all relative to what we accept as an "elementary" function. My point
is that the sum in the query cannot be done with simple factorial products
and ratios.

Cheers and Salud!

Henry