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A371928 T(n,k) is the total number of levels in all Dyck paths of semilength n containing exactly k path nodes; triangle T(n,k), n>=0, 1<=k<=n+1, read by rows.

%I A371928 #37 Apr 16 2024 18:02:03
%S A371928 1,1,1,1,3,1,3,5,6,1,8,15,13,11,1,23,44,43,29,20,1,71,134,138,106,62,
%T A371928 37,1,229,427,446,371,248,132,70,1,759,1408,1478,1275,941,571,283,135,
%U A371928 1,2566,4753,5017,4410,3437,2331,1310,611,264,1,8817,16321,17339,15458,12426,9027,5709,3002,1324,521,1
%N A371928 T(n,k) is the total number of levels in all Dyck paths of semilength n containing exactly k path nodes; triangle T(n,k), n>=0, 1<=k<=n+1, read by rows.
%C A371928 A Dyck path of semilength n has 2n+1 = A005408(n) nodes.
%H A371928 Alois P. Heinz, <a href="/A371928/b371928.txt">Rows n = 0..20, flattened</a>
%H A371928 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A371928 Sum_{k=1..n+1} k * T(n,k) = A001700(n) = A005408(n) * A000108(n).
%e A371928 In the A000108(3) = 5 Dyck paths of semilength 3 there are 3 levels with 1 node, 5 levels with 2 nodes, 6 levels with 3 nodes, and 1 level with 4 nodes.
%e A371928   1
%e A371928   2   /\      2           1           1
%e A371928   2  /  \     3  /\/\     3  /\       3    /\     3
%e A371928   2 /    \    2 /    \    3 /  \/\    3 /\/  \    4 /\/\/\    .
%e A371928   So row 3 is [3, 5, 6, 1].
%e A371928 Triangle T(n,k) begins:
%e A371928      1;
%e A371928      1,     1;
%e A371928      1,     3,     1;
%e A371928      3,     5,     6,     1;
%e A371928      8,    15,    13,    11,     1;
%e A371928     23,    44,    43,    29,    20,    1;
%e A371928     71,   134,   138,   106,    62,   37,    1;
%e A371928    229,   427,   446,   371,   248,  132,   70,    1;
%e A371928    759,  1408,  1478,  1275,   941,  571,  283,  135,    1;
%e A371928   2566,  4753,  5017,  4410,  3437, 2331, 1310,  611,  264,   1;
%e A371928   8817, 16321, 17339, 15458, 12426, 9027, 5709, 3002, 1324, 521, 1;
%e A371928   ...
%p A371928 g:= proc(x, y, p) (h-> `if`(x=0, add(z^coeff(h, z, i)
%p A371928           , i=0..degree(h)), b(x, y, h)))(p+z^y) end:
%p A371928 b:= proc(x, y, p) option remember; `if`(y+2<=x,
%p A371928       g(x-1, y+1, p), 0)+`if`(y>0, g(x-1, y-1, p), 0)
%p A371928     end:
%p A371928 T:= n-> (p-> seq(coeff(p, z, i), i=1..n+1))(g(2*n, 0$2)):
%p A371928 seq(T(n), n=0..10);
%Y A371928 Columns k=1-2 give: A152880, A371903.
%Y A371928 Row sums give A261003.
%Y A371928 T(n+1,n+1) gives A006127.
%Y A371928 Cf. A000108, A001700, A005408, A372014.
%K A371928 nonn,tabl
%O A371928 0,5
%A A371928 _Alois P. Heinz_, Apr 14 2024