OFFSET
0,6
COMMENTS
Row n is the inverse binomial transform of the n-th row of array A213276.
LINKS
Alois P. Heinz, Rows n = 0..20, flattened
FORMULA
T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A213276(n,k-i).
EXAMPLE
T(5,2) = 12: aaaab, aaaba, aaabb, aabaa, aabab, aabba, abaaa, abaab, ababa, baaaa, baaab, baaba.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 2;
0, 1, 3, 6;
0, 1, 7, 12, 24;
0, 1, 12, 35, 60, 120;
0, 1, 25, 87, 210, 360, 720;
0, 1, 44, 232, 609, 1470, 2520, 5040;
0, 1, 89, 599, 1961, 4872, 11760, 20160, 40320;
MATHEMATICA
g[l_, i_] := Module[{j}, If[l[[i]] < 1, Return[False], If[l[[i]] > 1, For[j = i + 1, j <= Length[l], j++, If[l[[i]] <= l[[j]], Return[False], If[l[[j]] > 0, Break[]]]]]]; True];
b[l_] := b[l] = If[Complement[l, {0}] == {}, 1, Sum[If[g[l, i], b[ReplacePart[l, i -> l[[i]] - 1]], 0], {i, 1, Length[l]}]];
h[n_, k_, m_, l_] := h[n, k, m, l] = If[n == 0 && k === 0, b[l], If[k == 0 || n > 0 && n < m, 0, Sum[h[n - j, k - 1, Max[m, j], Join[{j}, l]], {j, Max[1, m], n}] + h[n, k - 1, m, Join[{0}, l]]]];
A[n_, k_] := h[n, k, 0, {}];
T[n_, k_] := Sum[(-1)^i*Binomial[k, i]*A[n, k - i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 01 2022, after Alois P. Heinz in A213276 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, May 08 2015
STATUS
approved