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A372014 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.

%I A372014 #22 Apr 16 2024 19:05:28
%S A372014 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 A372014 84,64,34,16,6,1,113,180,208,181,111,52,22,7,1,296,457,520,485,344,
%U A372014 179,76,29,8,1,782,1195,1334,1273,1000,599,274,107,37,9,1
%N A372014 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 A372014 A Motzkin path of length n has n+1 nodes.
%H A372014 Alois P. Heinz, <a href="/A372014/b372014.txt">Rows n = 0..28, flattened</a>
%H A372014 Wikipedia, <a href="https://en.wikipedia.org/wiki/Motzkin_number">Motzkin number</a>
%F A372014 Sum_{k=1..n+1} k * T(n,k) = A005717(n+1) = (n+1) * A001006(n).
%e A372014 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 A372014   2  _     1        1
%e A372014   2 / \    3 /\_    3 _/\    4 ___    .
%e A372014   So row 3 is [2, 2, 2, 1].
%e A372014 Triangle T(n,k) begins:
%e A372014     1;
%e A372014     0,    1;
%e A372014     1,    1,    1;
%e A372014     2,    2,    2,    1;
%e A372014     4,    6,    4,    3,    1;
%e A372014     8,   14,   12,    7,    4,   1;
%e A372014    18,   32,   33,   21,   11,   5,   1;
%e A372014    44,   74,   84,   64,   34,  16,   6,   1;
%e A372014   113,  180,  208,  181,  111,  52,  22,   7,  1;
%e A372014   296,  457,  520,  485,  344, 179,  76,  29,  8, 1;
%e A372014   782, 1195, 1334, 1273, 1000, 599, 274, 107, 37, 9, 1;
%e A372014   ...
%p A372014 g:= proc(x, y, p) (h-> `if`(x=0, add(z^coeff(h, z, i)
%p A372014           , i=0..degree(h)), b(x, y, h)))(p+z^y) end:
%p A372014 b:= proc(x, y, p) option remember; `if`(y+2<=x, g(x-1, y+1, p), 0)
%p A372014       +`if`(y+1<=x, g(x-1, y, p), 0)+`if`(y>0, g(x-1, y-1, p), 0)
%p A372014     end:
%p A372014 T:= n-> (p-> seq(coeff(p, z, i), i=1..n+1))(g(n, 0$2)):
%p A372014 seq(T(n), n=0..10);
%Y A372014 Columns k=1-2 give: A088457, A051485.
%Y A372014 Row sums give A372033 = A001006 + A333498.
%Y A372014 Cf. A005717, A371928.
%K A372014 nonn,tabl
%O A372014 0,7
%A A372014 _Alois P. Heinz_, Apr 15 2024