A372014 - OEIS

%I #22 Apr 16 2024 19:05:28

%S 1,0,1,1,1,1,2,2,2,1,4,6,4,3,1,8,14,12,7,4,1,18,32,33,21,11,5,1,44,74,

%T 84,64,34,16,6,1,113,180,208,181,111,52,22,7,1,296,457,520,485,344,

%U 179,76,29,8,1,782,1195,1334,1273,1000,599,274,107,37,9,1

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

%C A Motzkin path of length n has n+1 nodes.

%H Alois P. Heinz, <a href="/A372014/b372014.txt">Rows n = 0..28, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Motzkin_number">Motzkin number</a>

%F Sum_{k=1..n+1} k * T(n,k) = A005717(n+1) = (n+1) * A001006(n).

%e In the A001006(3) = 4 Motzkin paths of length 3 there are 2 levels with 1 node, 2 levels with 2 nodes, 2 levels with 3 nodes, and 1 level with 4 nodes.

%e   2  _     1        1

%e   2 / \    3 /\_    3 _/\    4 ___    .

%e   So row 3 is [2, 2, 2, 1].

%e Triangle T(n,k) begins:

%e     1;

%e     0,    1;

%e     1,    1,    1;

%e     2,    2,    2,    1;

%e     4,    6,    4,    3,    1;

%e     8,   14,   12,    7,    4,   1;

%e    18,   32,   33,   21,   11,   5,   1;

%e    44,   74,   84,   64,   34,  16,   6,   1;

%e   113,  180,  208,  181,  111,  52,  22,   7,  1;

%e   296,  457,  520,  485,  344, 179,  76,  29,  8, 1;

%e   782, 1195, 1334, 1273, 1000, 599, 274, 107, 37, 9, 1;

%e   ...

%p g:= proc(x, y, p) (h-> `if`(x=0, add(z^coeff(h, z, i)

%p           , i=0..degree(h)), b(x, y, h)))(p+z^y) end:

%p b:= proc(x, y, p) option remember; `if`(y+2<=x, g(x-1, y+1, p), 0)

%p       +`if`(y+1<=x, g(x-1, y, p), 0)+`if`(y>0, g(x-1, y-1, p), 0)

%p     end:

%p T:= n-> (p-> seq(coeff(p, z, i), i=1..n+1))(g(n, 0$2)):

%p seq(T(n), n=0..10);

%Y Columns k=1-2 give: A088457, A051485.

%Y Row sums give A372033 = A001006 + A333498.

%Y Cf. A005717, A371928.

%K nonn,tabl

%O 0,7

%A _Alois P. Heinz_, Apr 15 2024