OFFSET
0,4
COMMENTS
T(n,k) = A000041(n) for n >= 0 and k >= n.
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
FORMULA
G.f. of column k: 1 + Sum_{j>0} x^j / Product_{i=0..k} (1-x^(i+j)).
EXAMPLE
T(6,0) = 4: [6], [3,3], [2,2,2], [1,1,1,1,1,1].
T(6,1) = 6: [6], [3,3], [2,1,1,1,1], [2,2,1,1], [2,2,2], [1,1,1,1,1,1].
T(6,2) = 9: [6], [4,2], [3,1,1,1], [3,2,1], [3,3], [2,1,1,1,1], [2,2,1,1], [2,2,2], [1,1,1,1,1,1].
Triangle begins:
1;
1, 1;
2, 2, 2;
2, 3, 3, 3;
3, 4, 5, 5, 5;
2, 5, 6, 7, 7, 7;
4, 6, 9, 10, 11, 11, 11;
2, 7, 10, 13, 14, 15, 15, 15;
MAPLE
b:= proc(n, i, k) option remember;
if n<0 or k<0 then 0
elif n=0 then 1
elif i<1 then 0
else b(n, i-1, k-1) +b(n-i, i, k)
fi
end:
T:= (n, k)-> `if`(n=0, 1, 0) +add(b(n-i, i, k), i=1..n):
seq(seq(T(n, k), k=0..n), n=0..20);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n < 0 || k < 0, 0, If[n == 0, 1, If[i < 1, 0, b[n, i-1, k-1] + b[n-i, i, k]]]]; t[n_, k_] := If[n == 0, 1, 0] + Sum[b[n-i, i, k], {i, 1, n}]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 30 2011
STATUS
approved