OFFSET
0,4
COMMENTS
T(n,k) = 1 for n >= 0 and k >= n.
In general, column k > 0 is asymptotic to c^(1/4) * r * exp(2*sqrt(c*n)) / (2*sqrt(Pi*(1-r)*(r + k*(1-r))) * n^(3/4)), where r is the smallest real root of the equation r^k + r = 1 and c = k*log(r)^2/2 + polylog(2, 1-r). - Vaclav Kotesovec, Jan 02 2016
LINKS
Alois P. Heinz, Rows n = 0..140
FORMULA
G.f. of column k: Sum_{j>=0} x^(j*((j-1)*k/2+1))/Product_{i=1..j} (1-x^i).
EXAMPLE
T(7,3) = 3: [7], [6,1], [5,2].
T(23,6) = 11: [23], [22,1], [21,2], [20,3], [19,4], [18,5], [17,6], [16,7], [15,8], [15,7,1], [14,8,1].
Triangle begins:
1;
1, 1;
2, 1, 1;
3, 2, 1, 1;
5, 2, 2, 1, 1;
7, 3, 2, 2, 1, 1;
11, 4, 3, 2, 2, 1, 1;
15, 5, 3, 3, 2, 2, 1, 1;
MAPLE
b:= proc(n, i, k) option remember;
if n<0 then 0
elif n=0 then 1
else add(b(n-i-j, i+j, k), j=k..n-i)
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, 0, If[n == 0, 1, Sum[b[n-i-j, i+j, k], {j, k, n-i}]]]; 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, Jan 19 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 29 2011
STATUS
approved