%I #20 Apr 13 2017 04:25:36

%S 1,1,4,1,9,2,18,3,41,8,2,89,16,4,185,34,10,388,57,10,810,113,30,6,

%T 1670,213,52,12,3435,396,104,28,7040,733,176,50,14360,1333,278,62,

%U 29226,2419,512,152,24,59347,4400,878,246,48,120229,7934,1492,458,108

%N Number T(n,k) of compositions of n such that k is the minimal distance between two identical parts; triangle T(n,k), n>=2, 1<=k<=floor((sqrt(8*n-7)-1)/2), read by rows.

%H Alois P. Heinz, <a href="/A261981/b261981.txt">Rows n = 2..55, flattened</a>

%F T(n,k) = A261960(n,k-1) - A261960(n,k).

%F T((n+1)*(n+2)/2+1,n+1) = A000142(n) for n>=0.

%e T(5,1) = 9: 311, 113, 221, 122, 2111, 1211, 1121, 1112, 11111.

%e T(5,2) = 2: 131, 212.

%e T(7,2) = 8: 151, 313, 232, 3121, 1213, 2131, 1312, 12121.

%e T(7,3) = 2: 1231, 1321.

%e Triangle T(n,k) begins:

%e n\k: 1 2 3 4 5

%e ---+---------------------------

%e 02 : 1;

%e 03 : 1;

%e 04 : 4, 1;

%e 05 : 9, 2;

%e 06 : 18, 3;

%e 07 : 41, 8, 2;

%e 08 : 89, 16, 4;

%e 09 : 185, 34, 10;

%e 10 : 388, 57, 10;

%e 11 : 810, 113, 30, 6;

%e 12 : 1670, 213, 52, 12;

%e 13 : 3435, 396, 104, 28;

%e 14 : 7040, 733, 176, 50;

%e 15 : 14360, 1333, 278, 62;

%e 16 : 29226, 2419, 512, 152, 24;

%p b:= proc(n, l) option remember;

%p `if`(n=0, 1, add(`if`(j in l, 0, b(n-j,

%p `if`(l=[], [], [subsop(1=NULL, l)[], j]))), j=1..n))

%p end:

%p T:= (n, k)-> b(n, [0$(k-1)])-b(n, [0$k]):

%p seq(seq(T(n, k), k=1..floor((sqrt(8*n-7)-1)/2)), n=2..20);

%t b[n_, l_] := b[n, l] = If[n == 0, 1, Sum[If[MemberQ[l, j], 0, b[n-j, If[l == {}, {}, Append[Rest[l], j]]]], {j, 1, n}]];

%t A[n_, k_] := b[n, Array[0&, Min[n, k]]];

%t T[n_, k_] := A[n, k-1] - A[n, k];

%t Table[T[n, k], {n, 2, 20}, {k, 1, Floor[(Sqrt[8*n-7]-1)/2]}] // Flatten (* _Jean-François Alcover_, Apr 13 2017, after _Alois P. Heinz_ *)

%Y Columns k=1-2 give: A261983, A261984.

%Y Row sums give A261982.

%Y Cf. A000142, A261960, A262191.

%K nonn,tabf

%O 2,3

%A _Alois P. Heinz_, Sep 07 2015