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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).

%I A238350 #29 Mar 14 2021 11:03:49
%S A238350 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 A238350 47,7,161,238,100,12,1,316,479,198,30,1,624,956,404,61,3,1235,1910,
%U A238350 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 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).
%D A238350 M. Archibald, A. Blecher and A. Knopfmacher, Fixed points in compositions and words, accepted by the Journal of Integer Sequences.
%H A238350 Joerg Arndt and Alois P. Heinz, <a href="/A238350/b238350.txt">Rows n = 0..500, flattened</a>
%H A238350 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 A238350 Sum_{k=0..A003056(n)} k * T(n,k) = A099036(n-1) for n>0.
%e A238350 Triangle T(n,k) begins:
%e A238350 00 :   1;
%e A238350 01 :   0,   1;
%e A238350 02 :   1,   1;
%e A238350 03 :   2,   1,   1;
%e A238350 04 :   3,   4,   1;
%e A238350 05 :   6,   7,   3;
%e A238350 06 :  11,  16,   4,  1;
%e A238350 07 :  22,  29,  12,  1;
%e A238350 08 :  42,  60,  23,  3;
%e A238350 09 :  82, 120,  47,  7;
%e A238350 10 : 161, 238, 100, 12, 1;
%e A238350 11 : 316, 479, 198, 30, 1;
%e A238350 12 : 624, 956, 404, 61, 3;
%e A238350      ...
%p A238350 b:= proc(n, i) option remember; `if`(n=0, 1, expand(
%p A238350        add(b(n-j, i+1)*`if`(i=j, x, 1), j=1..n)))
%p A238350     end:
%p A238350 T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n, 1)):
%p A238350 seq(T(n), n=0..20);
%t A238350 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 A238350 Columns k=0-10 give: A238351, A240736, A240737, A240738, A240739, A240740, A240741, A240742, A240743, A240744, A240745.
%Y A238350 Row sums are A011782.
%Y A238350 T(n*(n+3)/2,n) = A227682(n).
%Y A238350 Same as A238349 without the trailing zeros.
%Y A238350 Cf. A099036.
%K A238350 nonn,tabf
%O A238350 0,6
%A A238350 _Joerg Arndt_ and _Alois P. Heinz_, Feb 25 2014