%I #29 Mar 14 2021 11:03:49

%S 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,

%T 47,7,161,238,100,12,1,316,479,198,30,1,624,956,404,61,3,1235,1910,

%U 818,126,7,2449,3817,1652,258,16,4864,7633,3319,537,30,1,9676,15252,6686,1083,70,1,19267,30491,13426,2205

%N 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).

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

%H Joerg Arndt and Alois P. Heinz, <a href="/A238350/b238350.txt">Rows n = 0..500, flattened</a>

%H M. Archibald, A. Blecher, and A. Knopfmacher, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Blecher/arch14.html">Fixed Points in Compositions and Words</a>, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.

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

%e Triangle T(n,k) begins:

%e 00 : 1;

%e 01 : 0, 1;

%e 02 : 1, 1;

%e 03 : 2, 1, 1;

%e 04 : 3, 4, 1;

%e 05 : 6, 7, 3;

%e 06 : 11, 16, 4, 1;

%e 07 : 22, 29, 12, 1;

%e 08 : 42, 60, 23, 3;

%e 09 : 82, 120, 47, 7;

%e 10 : 161, 238, 100, 12, 1;

%e 11 : 316, 479, 198, 30, 1;

%e 12 : 624, 956, 404, 61, 3;

%e ...

%p b:= proc(n, i) option remember; `if`(n=0, 1, expand(

%p add(b(n-j, i+1)*`if`(i=j, x, 1), j=1..n)))

%p end:

%p T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n, 1)):

%p seq(T(n), n=0..20);

%t 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_ *)

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

%Y Row sums are A011782.

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

%Y Same as A238349 without the trailing zeros.

%Y Cf. A099036.

%K nonn,tabf

%O 0,6

%A _Joerg Arndt_ and _Alois P. Heinz_, Feb 25 2014