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

0,6

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.

Formula

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

Example

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;
     ...

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 *)

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.

Sequence in context: A089940 A331598 A123974 * A355754 A319844 A193736

Adjacent sequences: A238347 A238348 A238349 * A238351 A238352 A238353

Keyword

nonn,tabf

Author

Joerg Arndt and Alois P. Heinz, Feb 25 2014

Status

approved