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

1,4

Links

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

Formula

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

Example

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;

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 *)

Crossrefs

Cf. A048004.

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

Sequence in context: A355257 A129684 A247255 * A335262 A111924 A212880

Adjacent sequences: A105144 A105145 A105146 * A105148 A105149 A105150

Keyword

easy,nonn,tabl

Author

Vladeta Jovovic, Apr 10 2005

Status

approved