%I #18 Feb 08 2017 08:57:24

%S 1,0,1,3,1,1,4,4,2,1,5,6,6,3,1,12,13,12,9,4,1,21,23,25,21,13,5,1,36,

%T 42,46,46,34,18,6,1,43,68,88,92,80,52,24,7,1,88,119,152,180,172,132,

%U 76,31,8,1,133,197,267,330,352,304,208,107,39,9,1

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

%H Alois P. Heinz, <a href="/A262191/b262191.txt">Rows n = 2..20, flattened</a>

%e T(6,1) = 5: 33, 114, 411, 1122, 2211.

%e T(6,2) = 6: 141, 222, 1113, 1212, 2121, 3111.

%e T(6,3) = 6: 1131, 1221, 1311, 2112, 11112, 21111.

%e T(6,4) = 3: 11121, 11211, 12111.

%e T(6,5) = 1: 111111.

%e Triangle T(n,k) begins:

%e n\k: 1 2 3 4 5 6 7 8 9 10 11

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

%e 02 : 1;

%e 03 : 0, 1;

%e 04 : 3, 1, 1;

%e 05 : 4, 4, 2, 1;

%e 06 : 5, 6, 6, 3, 1;

%e 07 : 12, 13, 12, 9, 4, 1;

%e 08 : 21, 23, 25, 21, 13, 5, 1;

%e 09 : 36, 42, 46, 46, 34, 18, 6, 1;

%e 10 : 43, 68, 88, 92, 80, 52, 24, 7, 1;

%e 11 : 88, 119, 152, 180, 172, 132, 76, 31, 8, 1;

%e 12 : 133, 197, 267, 330, 352, 304, 208, 107, 39, 9, 1;

%p b:= proc(n, s, l) option remember; `if`(n=0, 1, add(

%p `if`(j in s, 0, b(n-j, s union {`if`(l=[], j, l[1])},

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

%p end:

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

%p seq(seq(T(n, k), k=1..n-1), n=2..14);

%t b[n_, s_, l_] := b[n, s, l] = If[n==0, 1, Sum[If[MemberQ[s, j], 0, b[n-j, s ~Union~ {If[l=={}, j, l[[1]]]}, If[l=={}, {}, Append[Rest[l], j]]]], {j, 1, n}]]; T[n_, k_] := b[n, {}, Array[0&, k]] - b[n, {}, Array[0&, k-1]]; Table[T[n, k], {n, 2, 14}, { k, 1, n-1}] // Flatten (* _Jean-François Alcover_, Feb 08 2017, translated from Maple *)

%Y Column k=1-5 gives A262192, A262194, A262196, A262197, A262200.

%Y Row sums give A261982.

%Y Cf. A261981.

%K nonn,tabl

%O 2,4

%A _Alois P. Heinz_, Sep 14 2015