A105147 Triangular array read by rows: T(n,k) = number of compositions of n having smallest part equal to k.

1, 1, 1, 3, 0, 1, 6, 1, 0, 1, 13, 2, 0, 0, 1, 27, 3, 1, 0, 0, 1, 56, 5, 2, 0, 0, 0, 1, 115, 9, 2, 1, 0, 0, 0, 1, 235, 15, 3, 2, 0, 0, 0, 0, 1, 478, 25, 5, 2, 1, 0, 0, 0, 0, 1, 969, 42, 8, 2, 2, 0, 0, 0, 0, 0, 1, 1959, 70, 12, 3, 2, 1, 0, 0, 0, 0, 0, 1, 3952, 116, 18, 5, 2, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Vladeta Jovovic, Apr 10 2005

Keywords

• easy
• nonn
• tabl

Examples

			1;
1,  1;
3,  0, 1;
6,  1, 0, 1;
13, 2, 0, 0, 1;
27, 3, 1, 0, 0, 1;
56, 5, 2, 0, 0, 0, 1;
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Cf. A048004.

Row sums give: A000079(n-1), columns k=1, 2 give: A099036(n-1), A200047. - Alois P. Heinz, Nov 13 2011

Programs

Maple

p:= (t, l)-> zip((x, y)->x+y, t, l, 0):
b:= proc(n) option remember; local j, t, h, m, s;
      t:= [0$(n-1), 1];
      for j to n-1 do
        h:= b(n-j);
        m:= nops(h);
        t:= p(p(t, [seq(h[i], i=1..min(j, m))]),
                   [0$(j-1), add(h[i], i=j+1..m)])
      od; t
    end:
T:= n-> b(n)[]:
seq(T(n), n=1..15); # Alois P. Heinz, Nov 13 2011

Mathematica

zip[f_, x_, y_, z_] := With[{m = Max[Length[x], Length[y]]}, Thread[f[PadRight[x, m, z], PadRight[y, m, z]]]]; p[t_, l_] := zip[Plus, t, l, 0]; b[n_] := b[n] = Module[{j, t, h, m, s}, t = Append[Array[0&, n-1], 1]; For[j = 1, j <= n-1 , j++, h = b[n-j]; m = Length[h]; t = p[p[t, h[[1 ;; Min[j, m]]]], Append[Array[0&, j-1], h[[Min[j, m]+1 ;; m]] // Total]]]; t]; Table[b[n], {n, 1, 15}] // Flatten (* Jean-François Alcover, Jan 29 2014, after Alois P. Heinz *)

Formula

G.f. for k-th column: (1-x)^2*x^k/((1-x-x^k)*(1-x-x^(k+1))).