OFFSET
0,6
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
FORMULA
T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A292804(n,k-i).
EXAMPLE
T(2,2) = 3: {ab}, {ba}, {a,b}.
T(3,2) = 12: {aab}, {aba}, {abb}, {baa}, {bab}, {bba}, {a,ab}, {a,ba}, {a,bb}, {aa,b}, {ab,b}, {b,ba}.
T(4,2) = 38: {aaab}, {aaba}, {aabb}, {abaa}, {abab}, {abba}, {abbb}, {baaa}, {baab}, {baba}, {babb}, {bbaa}, {bbab}, {bbba}, {a,aab}, {a,aba}, {a,abb}, {a,baa}, {a,bab}, {a,bba}, {a,bbb}, {aa,ab}, {aa,ba}, {aa,bb}, {aaa,b}, {aab,b}, {ab,ba}, {ab,bb}, {aba,b}, {abb,b}, {b,baa}, {b,bab}, {b,bba}, {ba,bb}, {a,aa,b}, {a,ab,b}, {a,b,ba}, {a,b,bb}.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 3;
0, 2, 12, 13;
0, 2, 38, 105, 73;
0, 3, 110, 588, 976, 501;
0, 4, 302, 2811, 8416, 9945, 4051;
0, 5, 806, 12354, 59488, 121710, 111396, 37633;
0, 6, 2109, 51543, 375698, 1185360, 1830822, 1366057, 394353;
MAPLE
h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))
end:
T:= (n, k)-> add((-1)^i*binomial(k, i)*h(n$2, k-i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);
MATHEMATICA
h[n_, i_, k_] := h[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[h[n-i*j, i-1, k]* Binomial[k^i, j], {j, 0, n/i}]]];
T[n_, k_] := Sum[(-1)^i Binomial[k, i] h[n, n, k-i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 05 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 20 2018
STATUS
approved