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

%I A105147 #17 Nov 19 2014 15:13:58
%S A105147 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,
%T A105147 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,
%U A105147 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
%N A105147 Triangular array read by rows: T(n,k) = number of compositions of n having smallest part equal to k.
%H A105147 Alois P. Heinz, <a href="/A105147/b105147.txt">Rows n = 1..141, flattened</a>
%F A105147 G.f. for k-th column: (1-x)^2*x^k/((1-x-x^k)*(1-x-x^(k+1))).
%e A105147 1;
%e A105147 1,  1;
%e A105147 3,  0, 1;
%e A105147 6,  1, 0, 1;
%e A105147 13, 2, 0, 0, 1;
%e A105147 27, 3, 1, 0, 0, 1;
%e A105147 56, 5, 2, 0, 0, 0, 1;
%p A105147 p:= (t, l)-> zip((x, y)->x+y, t, l, 0):
%p A105147 b:= proc(n) option remember; local j, t, h, m, s;
%p A105147       t:= [0$(n-1), 1];
%p A105147       for j to n-1 do
%p A105147         h:= b(n-j);
%p A105147         m:= nops(h);
%p A105147         t:= p(p(t, [seq(h[i], i=1..min(j, m))]),
%p A105147                    [0$(j-1), add(h[i], i=j+1..m)])
%p A105147       od; t
%p A105147     end:
%p A105147 T:= n-> b(n)[]:
%p A105147 seq(T(n), n=1..15); # _Alois P. Heinz_, Nov 13 2011
%t A105147 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_ *)
%Y A105147 Cf. A048004.
%Y A105147 Row sums give: A000079(n-1), columns k=1, 2 give: A099036(n-1), A200047. - _Alois P. Heinz_, Nov 13 2011
%K A105147 easy,nonn,tabl
%O A105147 1,4
%A A105147 _Vladeta Jovovic_, Apr 10 2005