A088457
Number of single nodes (exactly one node on that level) for all Motzkin paths of length n.
Original entry on oeis.org
1, 0, 1, 2, 4, 8, 18, 44, 113, 296, 782, 2076, 5538, 14856, 40100, 108936, 297793, 818832, 2263481, 6286498, 17532707, 49077268, 137821247, 388150322, 1095980561, 3101840232, 8797579789, 25001305410, 71179961918, 203000438544, 579876376729, 1658948939262
Offset: 0
[0,0,0,1,0], [0,0,1,0,0], [0,1,0,0,0], [0,1,2,1,0] are the a(4) = 4 sequences.
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b:= proc(x, y, h, c) option remember; `if`(y<0 or y>x, 0,
`if`(x=0, c, add(b(x-1, y-i, max(h, y),
`if`(h=y, 0, `if`(h b(n, 0$2, 1):
seq(a(n), n=0..31); # Alois P. Heinz, Jul 25 2023
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b[x_, y_, h_, c_] := b[x, y, h, c] = If[y<0 || y>x, 0, If[x == 0, c, Sum[b[x-1, y-i, Max[h, y], If[h == y, 0, If[h < y, 1, c]]], {i, -1, 1}]]];
a[n_] := b[n, 0, 0, 1];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Oct 23 2023, after Alois P. Heinz *)
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{a(n)=local(p0, p1, p2); if(n<0, 0, p1=1; polcoeff(sum(i=0, n, if(p2=(1-x)*p1-x^2*p0, p0=p1; p1=p2; (x^i/p0)^2), x*O(x^n)), 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.
Original entry on oeis.org
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, 84, 64, 34, 16, 6, 1, 113, 180, 208, 181, 111, 52, 22, 7, 1, 296, 457, 520, 485, 344, 179, 76, 29, 8, 1, 782, 1195, 1334, 1273, 1000, 599, 274, 107, 37, 9, 1
Offset: 0
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.
2 _ 1 1
2 / \ 3 /\_ 3 _/\ 4 ___ .
So row 3 is [2, 2, 2, 1].
Triangle T(n,k) begins:
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, 84, 64, 34, 16, 6, 1;
113, 180, 208, 181, 111, 52, 22, 7, 1;
296, 457, 520, 485, 344, 179, 76, 29, 8, 1;
782, 1195, 1334, 1273, 1000, 599, 274, 107, 37, 9, 1;
...
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g:= proc(x, y, p) (h-> `if`(x=0, add(z^coeff(h, z, i)
, i=0..degree(h)), b(x, y, h)))(p+z^y) end:
b:= proc(x, y, p) option remember; `if`(y+2<=x, g(x-1, y+1, p), 0)
+`if`(y+1<=x, g(x-1, y, p), 0)+`if`(y>0, g(x-1, y-1, p), 0)
end:
T:= n-> (p-> seq(coeff(p, z, i), i=1..n+1))(g(n, 0$2)):
seq(T(n), n=0..10);
A371903
Total number of levels in all Dyck paths of semilength n containing exactly 2 path nodes.
Original entry on oeis.org
0, 1, 3, 5, 15, 44, 134, 427, 1408, 4753, 16321, 56812, 200046, 711425, 2551886, 9222147, 33544682, 122712465, 451169747, 1666248405, 6178586630, 22994275870, 85859249486, 321562877934, 1207665205311, 4547078084804
Offset: 0
a(3) = 3 + 2 + 0 + 0 + 0 = 5:
1
_2 /\ _2 1 1
_2 / \ 3 /\/\ 3 /\ 3 /\ 3
_2 / \ _2 / \ 3 / \/\ 3 /\/ \ 4 /\/\/\ .
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g:= proc(x, y, p) (h-> `if`(x=0, add(`if`(coeff(h, z, i)=2, 1, 0),
i=0..degree(h)), b(x, y, h)))(p+`if`(coeff(p, z, y)<3, z^y, 0))
end:
b:= proc(x, y, p) option remember; `if`(y+2<=x,
g(x-1, y+1, p), 0)+`if`(y>0, g(x-1, y-1, p), 0)
end:
a:= n-> g(2*n, 0$2):
seq(a(n), n=0..18);
Showing 1-3 of 3 results.
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