OFFSET
0,6
COMMENTS
T(n,k) is defined for n,k >= 0. The triangle contains terms with k <= n. T(n,k) = T(n,n) = A006128(n) for k >= n.
For fixed k > 0, T(n,k) ~ 3^(1/4) * log(k+1) * exp(Pi*sqrt(2*k*n/(3*(k+1)))) / (Pi * (8*k*(k+1)*n)^(1/4)). - Vaclav Kotesovec, Oct 18 2018
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
FORMULA
T(n,k) = Sum_{i=0..k} A213177(n,i).
EXAMPLE
T(6,2) = 17: [6], [5,1], [4,2], [3,3], [4,1,1], [3,2,1], [2,2,1,1].
Triangle T(n,k) begins:
0;
0, 1;
0, 1, 3;
0, 3, 3, 6;
0, 3, 8, 8, 12;
0, 5, 11, 15, 15, 20;
0, 8, 17, 24, 29, 29, 35;
0, 10, 23, 36, 41, 47, 47, 54;
0, 13, 36, 50, 65, 71, 78, 78, 86;
...
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
end:
T:= (n, k)-> b(n, n, k)[2]:
seq(seq(T(n, k), k=0..n), n=0..12);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[b[n-i*j, i-1, k] /. l_List :> {l[[1]], l[[2]] + l[[1]]*j}, {j, 0, Min[n/i, k]}]]]; T[n_, k_] := b[n, n, k][[2]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jan 23 2013
STATUS
approved