proposed
approved
editing
b[n_, i_] := If[n == 0, {5}, If[i<1, {}, Table[Map[Function[x, x*Binomial[ 4^i + j - 1, j]], b[n - i*j, i - 1]], {j, 0, n/i}]] // Flatten];
T[n_] := b[n, n];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 17 2018, after Alois P. Heinz *)
reviewed
Row sums give 5*A144068.
Row lengths give A000041.
b:= (n, i)-> `if`(n=0, [5], `if`(i<1, [], [seq(map(x-> x*
binomial(4^i+j-1, j), b(n-i*j, i-1))[], j=0..n/i)])):
T:= n-> b(n$2)[]:
seq(T(n), n=0..10); # Alois P. Heinz, Jan 14 2018
