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 *)
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);
Original entry on oeis.org
1, 2, 5, 15, 48, 160, 549, 1924, 6851, 24700, 89945, 330239, 1220884, 4540128, 16968958, 63701573, 240059998, 907760348, 3443048256, 13094812968, 49925646786, 190772846082, 730451716847, 2802033270234, 10767028435468, 41438212118088, 159711845145544, 616393788920923, 2381898673172602
Offset: 0
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# Maple code for Equations (1) and (2) of Dershowitz and Rinderknecht (2015).
H:=proc(n,h) local b,k; b:=binomial; add(b(2*n,n+1-k*h)-2*b(2*n,n-k*h)+b(2*n,n-1-k*h),k=1..n+1); end;
S1:=n->add(H(n,h),h=1..n+1); [seq(S1(n),n=0..30)];
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b[x_, y_, h_] := b[x, y, h] = If[x == 0, h, Sum[If[x+j > y, b[x-1, y-j, Max[h, y-j]], 0], {j, Range[-1, Min[1, y]]~Complement~{0}}]];
a[n_] := b[2n, 0, 0] + CatalanNumber[n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 15 2023, after Alois P. Heinz in A136439 *)
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 1-4 of 4 results.
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