A152879 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k peaks of maximum height (1 <= k <= n).
1, 1, 1, 3, 1, 1, 8, 4, 1, 1, 23, 12, 5, 1, 1, 71, 36, 17, 6, 1, 1, 229, 114, 54, 23, 7, 1, 1, 759, 377, 176, 78, 30, 8, 1, 1, 2566, 1279, 596, 263, 109, 38, 9, 1, 1, 8817, 4408, 2070, 912, 382, 148, 47, 10, 1, 1, 30717, 15375, 7289, 3240, 1358, 541, 196, 57, 11, 1, 1, 108278
Offset: 1
Examples
T(4,2)=4 because we have UU(UD)(UD)DD, U(UD)DU(UD)D, U(UD)(UD)DUD and UDU(UD)(UD)D, where U=(1,1), D=(1,-1), with the peaks of maximum height shown between parentheses. Triangle starts: 1; 1, 1; 3, 1, 1; 8, 4, 1, 1; 23, 12, 5, 1, 1; 71, 36, 17, 6, 1, 1; ...
Links
- Alois P. Heinz, Rows n = 1..150, flattened
Programs
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Maple
f[0] := 1: f[1] := 1: for i from 2 to 20 do f[i] := sort(expand(f[i-1]-z*f[i-2])) end do: G := sum(t*z^j/(f[j]*(f[j]-t*z*f[j-1])), j = 1 .. 20): Gser := simplify(series(G, z = 0, 17)): for n to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n to 12 do seq(coeff(P[n], t, k), k = 1 .. n) end do; # yields sequence in triangular form
Formula
G.f. = G(t,z) = Sum_{j>=1}tz^j/(f(j)(f(j)-tzf(j-1))), 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 (Sergi Elizalde).
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