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A105147 Triangular array read by rows: T(n,k) = number of compositions of n having smallest part equal to k. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A102765 A129684 A247255 * A111924 A212880 A211510

Adjacent sequences:  A105144 A105145 A105146 * A105148 A105149 A105150

KEYWORD

easy,nonn,tabl

AUTHOR

Vladeta Jovovic, Apr 10 2005

STATUS

approved

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Last modified March 27 22:02 EDT 2017. Contains 284182 sequences.