OFFSET
1,2
COMMENTS
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
EXAMPLE
Triangle begins:
1;
2, 1;
4, 0, 1;
7, 2, 0, 1;
12, 1, 0, 0, 1;
19, 4, 2, 0, 0, 1;
30, 3, 1, 0, 0, 0, 1;
45, 8, 1, 2, 0, 0, 0, 1;
67, 7, 4, 1, 0, 0, 0, 0, 1;
97, 15, 3, 1, 2, 0, 0, 0, 0, 1;
139, 15, 4, 1, 1, 0, 0, 0, 0, 0, 1;
195, 27, 8, 4, 1, 2, 0, 0, 0, 0, 0, 1;
272, 29, 8, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1;
...
The partitions of 6 with the smallest part in brackets are
..........................
. [6]
..........................
. [3]+[3]
..........................
. 4 +[2]
. [2]+[2]+[2]
..........................
. 5 +[1]
. 3 + 2 +[1]
. 4 +[1]+[1]
. 2 + 2 +[1]+[1]
. 3 +[1]+[1]+[1]
. 2 +[1]+[1]+[1]+[1]
. [1]+[1]+[1]+[1]+[1]+[1]
..........................
There are 19 smallest parts of size 1. Also there are four smallest parts of size 2. Also there are two smallest parts of size 3. There are no smallest part of size 4 or 5. Finally there is only one smallest part of size 6. So row 6 gives 19, 4, 2, 0, 0, 1. The sum of row 6 is 19+4+2+0+0+1 = A092269(6) = 26.
MAPLE
b:= proc(n, i) option remember; local j, r; if n=0 or i<1 then 0
else `if`(irem(n, i, 'r')=0, [0$(i-1), r], []); for j from 0
to n/i do zip((x, y)->x+y, %, [b(n-i*j, i-1)], 0) od; %[] fi
end:
T:= n-> b(n, n):
seq(T(n), n=1..20); # Alois P. Heinz, Jan 20 2013
MATHEMATICA
b[n_, i_] := b[n, i] = Module[{j, q, r, pc}, If [n == 0 || i<1, 0, {q, r} = QuotientRemainder[n, i]; pc = If[r == 0, Append[Array[0&, i-1], q], {}]; For[j = 0, j <= n/i, j++, pc = Plus @@ PadRight[{pc, b[n-i*j, i-1]}]]; pc]]; T[n_] := b[n, n]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Jan 19 2013
STATUS
approved