@dharr Greetings from The On-Line Encyclopedia of Integer Sequences! https://oeis.org/

> lookup 1 2 4 5 8 12 14 16 28 32 37 64 94 106 128 144 232 256 289 320 512

a(n) = 1, 2, 4, 5, 8, 12, 14, 16, 28, 32, 37, 64, 94, 106, 128, 144, 232, 256, 289, 320, 512

# Direct matches

These sequences match the query directly.

oeis.org/search?q=1,2,4,5,8,12,14,16,28,32,37,64,94,106,128,144,232,256,289,320,512

oeis.org/A035001 Sorted elements of table (A035002) of a(m,n) =

 sum(a(m-k,n), k=1..m-1)+sum(a(m,n-k), k=1..n-1).

    <1, 2, 4, 5, 8, 12, 14, 16, 28, 32, 37, 64, 94, 106, 128, 144, 232, 256, 289, 320, 512>, 560, 704, 760, 838, 1024, 1328, 1536, 1944, 2048, 2329, 3104, 3328, 4096, 4864, 6266, 6802, 7168, 8192, 11952,1 5360, 16384, 16428, 19149, 28928, 32768, 37120, 42168

# Listtoalgeq

The gfun package's listtoalgeq function identified these possible algebraicq equations satisfied by the generating function for the query sequence. Warning! These may only be approximations!

    lgdegf 1024-5120*a(n)+11520*a(n)^2-15360*a(n)^3+13440*a(n)^4-8064*a(n)^5+3360*a(n)^6-960*a(n)^7+180*a(n)^8-20*a(n)^9+a(n)^10

@dharr To reproduce send email to address superseeker@oeis.org with empty subject and following body content:

lookup 1 2 4 5 8 12 14 16 28 32 37 64 94 106 128 144 232 256 289 320 512 560 704 760 838 1024 1328 1536 1944 2048 2329 3104 3328 4096 4864 6266 6802 7168 8192 11952 15360 16384 16428 19149 28928 32768

From: <israel@math.ubc.ca>
Subject: Re: Identity for Sqrt[Pi] - known, corollary, trivial or interesting?
To: Alexander P-sky <apovolot@gmail.com>

Maple simplifies the right hand side of the expression to sqrt(Pi). 
It seems to be true for arbitrary j,k,l,m.

Cheers,
Robert
--------
From: Victor K, PracticalWolf.com
https://community.wolfram.com/groups/-/m/t/2789785

FullSimplify[(1/(2^
       j) ((k*Gamma[5 + 2 j] Gamma[
          1 + l] HypergeometricPFQ[{1, 5/2 + j, 3 + j}, {3 + j + l/2, 
           7/2 + j + l/2}, -1])/
       Gamma[6 + 2 j + 
         l] + ((k + m) Gamma[7 + 2 j] Gamma[
          1 + l] HypergeometricPFQ[{1, 7/2 + j, 4 + j}, {4 + j + l/2, 
           9/2 + j + l/2}, -1])/Gamma[8 + 2 j + l]))/(2^(-5 - 3 j - 
       l) Gamma[5 + 2 j] Gamma[
     1 + l] (k HypergeometricPFQRegularized[{1, 5/2 + j, 
         3 + j}, {3 + j + l/2, 7/2 + j + l/2}, -1] + 
      1/2 (3 + j) (5 + 2 j) (k + m) HypergeometricPFQRegularized[{1, 
         7/2 + j, 4 + j}, {4 + j + l/2, 9/2 + j + l/2}, -1])),
 {j \[Element] Integers, k \[Element] Integers, 
  l \[Element] Integers, m \[Element] Integers}]
returns Sqrt[Pi], so your hypothesis seems to be confirmed by Mathematica. I'm not sure it's easy to produce a sequence of simplifying steps to "prove" that for humans.