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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).
16
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
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
KEYWORD
nonn,tabf
AUTHOR
Joerg Arndt and Alois P. Heinz, Feb 25 2014
STATUS
approved