OFFSET
0,3
COMMENTS
T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero.
LINKS
Alois P. Heinz, Columns k = 0..40, flattened
Wikipedia, Partition (number theory)
FORMULA
EXAMPLE
T(3,2) = 2: 2a1b, 2b1a.
T(3,3) = 5: 3abc, 2ab1c, 2ac1b, 2bc1a, 111abc.
Triangle T(n,k) begins:
1;
1;
2;
2, 5;
1, 9, 13;
9, 44, 42;
10, 96, 225, 150;
9, 152, 680, 1098, 576;
3, 155, 1350, 4155, 5201, 2266;
124, 2180, 11730, 26642, 26904, 9966;
140, 3751, 30300, 106281, 182000, 149832, 47466;
...
MAPLE
g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))
end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), n=k..g(k)), k=0..6);
MATHEMATICA
g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[n - 1]] ;
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = i j}, b[n - t, Min[n - t, i - 1], k]*Binomial[k, t]], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {n, k, g[k]}], {k, 0, 6}] // Flatten (* Jean-François Alcover, Mar 12 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Sep 12 2019
STATUS
approved