A152880
Number of Dyck paths of semilength n having exactly one peak of maximum height.
Original entry on oeis.org
1, 1, 3, 8, 23, 71, 229, 759, 2566, 8817, 30717, 108278, 385509, 1384262, 5006925, 18225400, 66711769, 245400354, 906711758, 3363516354, 12522302087, 46773419089, 175232388955, 658295899526, 2479268126762, 9359152696924, 35406650450001, 134215036793130
Offset: 1
a(3)=3 because we have UU(UD)DD, UDU(UD)D, U(UD)DUD, where U=(1,1), D=(1,-1), with the peak of maximum height shown between parentheses; the path UUDUDD does not qualify because it has two peaks of maximum height.
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f[0] := 1: f[1] := 1: for i from 2 to 35 do f[i] := sort(expand(f[i-1]-z*f[i-2])) end do; g := sum(z^j/f[j]^2, j = 1 .. 34): gser := series(g, z = 0, 30): seq(coeff(gser, z, n), n = 1 .. 27);
# second Maple program:
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(2*n, 0$3):
seq(a(n), n=1..28); # 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[hJean-François Alcover, Sep 17 2024, after Alois P. Heinz *)
A364030
Number of Dyck paths of semilength 2n having exactly n peaks of maximal height.
Original entry on oeis.org
1, 1, 4, 17, 78, 382, 1975, 10663, 59526, 340802, 1988575, 11771619, 70465201, 425572175, 2589083577, 15849320663, 97548380006, 603277351699, 3747168602548, 23367880883262, 146262143795248, 918597886652632, 5787577778959613, 36572319110701681, 231742772456205071
Offset: 0
a(0) = 1: (), the empty Dyck path.
a(1) = 1: //^\\.
a(2) = 4: ///^\/^\\\, //^\\//^\\, //^\/^\\/\, /\//^\/^\\.
A364386
Triangle T(n,k) read by rows: the number of Motzkin paths of length n that have k nodes at their peak level, 1 <= k <= n+1.
Original entry on oeis.org
1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 8, 7, 4, 1, 0, 1, 18, 15, 11, 5, 1, 0, 1, 44, 33, 26, 16, 6, 1, 0, 1, 113, 78, 59, 42, 22, 7, 1, 0, 1, 296, 197, 138, 101, 64, 29, 8, 1, 0, 1, 782, 518, 342, 240, 165, 93, 37, 9, 1, 0, 1, 2076, 1388, 892, 590, 406, 258, 130, 46, 10, 1, 0, 1
Offset: 0
Example for 9 paths of length n=4: UUDD (k=1 at level 2), UHHD (k=3 at level 1), UHDH (k=2 at level 1), UDUD (k=2 at level 1), UDHH (k=1 at level 1), HUHD (k=2 at level 1), HUDH (k=1 at level 1), HHUD (k=1 at level 1), HHHH (k=5 at level 0). So k=1 appears 4 times, k=2 3 times, k=3 once, k=4 never, k=5 once.
The triangle starts:
1
0, 1
1, 0, 1
2, 1, 0, 1
4, 3, 1, 0, 1
8, 7, 4, 1, 0, 1
18, 15, 11, 5, 1, 0, 1
44, 33, 26, 16, 6, 1, 0, 1
113, 78, 59, 42, 22, 7, 1, 0, 1
296, 197, 138, 101, 64, 29, 8, 1, 0, 1
782, 518, 342, 240, 165, 93, 37, 9, 1, 0, 1
2076, 1388, 892, 590, 406, 258, 130, 46, 10, 1, 0, 1
5538, 3747, 2401, 1522, 1005, 665, 388, 176, 56, 11, 1, 0, 1
14856, 10147, 6560, 4085, 2576, 1680, 1054, 564, 232, 67, 12, 1, 0, 1
...
Cf.
A152879 (equivalent for Dyck paths).
Showing 1-3 of 3 results.
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