A262191 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.

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, 42, 46, 46, 34, 18, 6, 1, 43, 68, 88, 92, 80, 52, 24, 7, 1, 88, 119, 152, 180, 172, 132, 76, 31, 8, 1, 133, 197, 267, 330, 352, 304, 208, 107, 39, 9, 1

Offset: 2

Author

Alois P. Heinz, Sep 14 2015

Keywords

• nonn
• tabl

Examples

			T(6,1) = 5: 33, 114, 411, 1122, 2211.
T(6,2) = 6: 141, 222, 1113, 1212, 2121, 3111.
T(6,3) = 6: 1131, 1221, 1311, 2112, 11112, 21111.
T(6,4) = 3: 11121, 11211, 12111.
T(6,5) = 1: 111111.
Triangle T(n,k) begins:
n\k:   1    2    3    4    5    6    7    8   9  10  11
---+----------------------------------------------------
02 :   1;
03 :   0,   1;
04 :   3,   1,   1;
05 :   4,   4,   2,   1;
06 :   5,   6,   6,   3,   1;
07 :  12,  13,  12,   9,   4,   1;
08 :  21,  23,  25,  21,  13,   5,   1;
09 :  36,  42,  46,  46,  34,  18,   6,   1;
10 :  43,  68,  88,  92,  80,  52,  24,   7,  1;
11 :  88, 119, 152, 180, 172, 132,  76,  31,  8,  1;
12 : 133, 197, 267, 330, 352, 304, 208, 107, 39,  9,  1;
		

Links

• Alois P. Heinz, Rows n = 2..20, flattened

Crossrefs

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

Row sums give A261982.

Cf. A261981.

Programs

Maple

b:= proc(n, s, l) option remember; `if`(n=0, 1, add(
      `if`(j in s, 0, b(n-j, s union {`if`(l=[], j, l[1])},
      `if`(l=[], [], [subsop(1=NULL, l)[], j]))), j=1..n))
    end:
T:= (n, k)-> b(n, {}, [0$k]) -b(n, {}, [0$(k-1)]):
seq(seq(T(n, k), k=1..n-1), n=2..14);

Mathematica

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 *)