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A152880
Number of Dyck paths of semilength n having exactly one peak of maximum height.
5
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
1,3
COMMENTS
Also number of peaks of maximum height in all Dyck paths of semilength n-1. Example: a(3)=3 because in (UD)(UD) and U(UD)D we have three peaks of maximum height (shown between parentheses).
FORMULA
G.f.: g(z) = Sum_{j>=1} z^j/f(j)^2, where the f(j)'s are the Fibonacci polynomials (in z) defined by f(0)=f(1)=1, f(j)=f(j-1)-zf(j-2), j>=2.
a(n) = A152879(n,1).
a(n) = Sum_{k=1..n} k*A152879(n-1,k).
EXAMPLE
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.
MAPLE
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<y, 1, c))), i=[1, -1])))
end:
a:= n-> b(2*n, 0$3):
seq(a(n), n=1..28); # Alois P. Heinz, Jul 25 2023
MATHEMATICA
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[2*n, 0, 0, 0];
Table[a[n], {n, 1, 28}] (* Jean-François Alcover, Sep 17 2024, after Alois P. Heinz *)
CROSSREFS
Column k=1 of A371928.
Sequence in context: A148776 A353067 A127385 * A259441 A176605 A080410
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jan 02 2009
STATUS
approved