OFFSET
0,5
COMMENTS
T(n,k) is defined for all n>=0 and k>=0. The triangle contains only elements with 0 <= k <= A003056(n). T(n,k) = 0 for k > A003056(n).
For fixed k>=1, T(n,k) ~ exp(sqrt(2*(Pi^2 - 6*polylog(2, 1-k))*n/3)) * sqrt(Pi^2 - 6*polylog(2, 1-k)) / (4*k!*sqrt(3*k)*Pi*n). - Vaclav Kotesovec, Sep 18 2019
LINKS
Alois P. Heinz, Rows n = 0..1000, flattened
FORMULA
EXAMPLE
T(6,1) = 11: 111111a, 2a1111a, 22a11a, 222a, 3a111a, 3a2a1a, 33a, 4a11a, 4a2a, 5a1a, 6a.
T(6,2) = 9: 2a1111b, 22a11b, 3a111b, 3a2a1b, 3a2b1a, 3a2b1b, 4a11b, 4a2b, 5a1b.
T(6,3) = 1: 3a2b1c.
Triangle T(n,k) begins:
1;
0, 1;
0, 2;
0, 3, 1;
0, 5, 2;
0, 7, 5;
0, 11, 9, 1;
0, 15, 17, 2;
0, 22, 28, 5;
0, 30, 47, 10;
0, 42, 74, 21, 1;
0, 56, 116, 37, 2;
0, 77, 175, 67, 5;
...
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
(t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!:
seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..20);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k]], {j, 1, n/i}] k + b[n, i - 1, k]]];
T[n_, k_] := Sum[b[n, n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]/k!;
Table[Table[T[n, k], {k, 0, Floor[(Sqrt[1 + 8n] - 1)/2]}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
AUTHOR
Alois P. Heinz, Aug 27 2019
STATUS
approved