A238350 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with k parts p at position p (fixed points), n>=0, 0<=k<=A003056(n).

1, 0, 1, 1, 1, 2, 1, 1, 3, 4, 1, 6, 7, 3, 11, 16, 4, 1, 22, 29, 12, 1, 42, 60, 23, 3, 82, 120, 47, 7, 161, 238, 100, 12, 1, 316, 479, 198, 30, 1, 624, 956, 404, 61, 3, 1235, 1910, 818, 126, 7, 2449, 3817, 1652, 258, 16, 4864, 7633, 3319, 537, 30, 1, 9676, 15252, 6686, 1083, 70, 1, 19267, 30491, 13426, 2205

Offset: 0

Author

Joerg Arndt and Alois P. Heinz, Feb 25 2014

Keywords

• nonn
• tabf

Examples

			Triangle T(n,k) begins:
00 :   1;
01 :   0,   1;
02 :   1,   1;
03 :   2,   1,   1;
04 :   3,   4,   1;
05 :   6,   7,   3;
06 :  11,  16,   4,  1;
07 :  22,  29,  12,  1;
08 :  42,  60,  23,  3;
09 :  82, 120,  47,  7;
10 : 161, 238, 100, 12, 1;
11 : 316, 479, 198, 30, 1;
12 : 624, 956, 404, 61, 3;
     ...
		

References

• M. Archibald, A. Blecher and A. Knopfmacher, Fixed points in compositions and words, accepted by the Journal of Integer Sequences.

Links

• Joerg Arndt and Alois P. Heinz, Rows n = 0..500, flattened
• M. Archibald, A. Blecher, and A. Knopfmacher, Fixed Points in Compositions and Words, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.

Crossrefs

Columns k=0-10 give: A238351, A240736, A240737, A240738, A240739, A240740, A240741, A240742, A240743, A240744, A240745.

Row sums are A011782.

T(n*(n+3)/2,n) = A227682(n).

Same as A238349 without the trailing zeros.

Cf. A099036.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, expand(
       add(b(n-j, i+1)*`if`(i=j, x, 1), j=1..n)))
    end:
T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n, 1)):
seq(T(n), n=0..20);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, Expand[Sum[b[n-j, i+1]*If[i == j, x, 1], {j, 1, n}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 1]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Alois P. Heinz *)

Formula

Sum_{k=0..A003056(n)} k * T(n,k) = A099036(n-1) for n>0.