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))}
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.
Original entry on oeis.org
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, 37, 1, 229, 427, 446, 371, 248, 132, 70, 1, 759, 1408, 1478, 1275, 941, 571, 283, 135, 1, 2566, 4753, 5017, 4410, 3437, 2331, 1310, 611, 264, 1, 8817, 16321, 17339, 15458, 12426, 9027, 5709, 3002, 1324, 521, 1
Offset: 0
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.
1
2 /\ 2 1 1
2 / \ 3 /\/\ 3 /\ 3 /\ 3
2 / \ 2 / \ 3 / \/\ 3 /\/ \ 4 /\/\/\ .
So row 3 is [3, 5, 6, 1].
Triangle T(n,k) begins:
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, 37, 1;
229, 427, 446, 371, 248, 132, 70, 1;
759, 1408, 1478, 1275, 941, 571, 283, 135, 1;
2566, 4753, 5017, 4410, 3437, 2331, 1310, 611, 264, 1;
8817, 16321, 17339, 15458, 12426, 9027, 5709, 3002, 1324, 521, 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>0, g(x-1, y-1, p), 0)
end:
T:= n-> (p-> seq(coeff(p, z, i), i=1..n+1))(g(2*n, 0$2)):
seq(T(n), n=0..10);
A051485
Number of double nodes (exactly two nodes on that level) for all Motzkin paths of length n.
Original entry on oeis.org
0, 1, 1, 2, 6, 14, 32, 74, 180, 457, 1195, 3177, 8526, 23018, 62441, 170153, 465791, 1280956, 3538618, 9817619, 27348480, 76467497, 214532805, 603732396, 1703728554, 4819990947, 13667248631, 38834528740, 110556072877, 315290709729, 900635841754, 2576615923655, 7381956798465, 21177682172332, 60832837964492
Offset: 0
Of the 9 Motzkin paths of length 4 the following 5 have a total of 6 double nodes:
|1......|
|2../\..|3..__..|2.._...|2..._..|2......|
|2./..\.|2./..\.|3./.\_.|3._/.\.|3./\/\.|
A372033
The total number of levels visited by all Motzkin paths of length n.
Original entry on oeis.org
1, 1, 3, 7, 18, 46, 121, 323, 875, 2395, 6611, 18371, 51337, 144145, 406420, 1150126, 3265412, 9298372, 26547710, 75978322, 217921336, 626287520, 1803176384, 5200298000, 15020569818, 43447201226, 125837214564, 364911724264, 1059404265599, 3078918594707
Offset: 0
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b:= proc(x, y, h) option remember; `if`(x=0, h+1, add(
b(x-1, y+j, max(h, y)), j=-min(1, y)..min(1, x-y-1)))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..35);
Showing 1-4 of 4 results.
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