A143953 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k peaks in their peak plateaux (0<=k<=n-1). A peak plateau is a run of consecutive peaks that is preceded by an upstep and followed by a down step; a peak consists of an upstep followed by a downstep.
1, 1, 1, 1, 1, 3, 1, 1, 8, 4, 1, 1, 21, 14, 5, 1, 1, 55, 48, 21, 6, 1, 1, 144, 162, 85, 29, 7, 1, 1, 377, 537, 335, 133, 38, 8, 1, 1, 987, 1748, 1286, 589, 193, 48, 9, 1, 1, 2584, 5594, 4815, 2526, 940, 266, 59, 10, 1, 1, 6765, 17629, 17619, 10518, 4413, 1405, 353, 71, 11, 1
Offset: 0
Examples
T(4,2)=4 because we have UDU(UDUD)D, U(UDUD)DUD, U(UD)DU(UD)D and UU(UDUD)DD (the peaks in the peak plateaux are shown between parentheses). The triangle starts: 1; 1; 1,1; 1,3,1; 1,8,4,1; 1,21,14,5,1; 1,55,48,21,6,1;
Programs
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Maple
C:=proc(z) options operator, arrow: (1/2-(1/2)*sqrt(1-4*z))/z end proc: G:=(1-z)*(1-t*z)*C(z*(1-z)^2*(1-t*z)^2/(1-z+z^2-t*z)^2)/(1-z+z^2-t*z): Gser:= simplify(series(G,z=0,15)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) end do: 1; for n to 11 do seq(coeff(P[n],t,j),j=0..n-1) end do; # yields sequence in triangular form
Formula
T(n,1) = A001906(n-1) = Fibonacci(2*n-2).
Sum_{k=0..n-1} k*T(n,k) = A143954(n).
The g.f. G=G(t,z) satisfies z(1-z)(1-tz)G^2-(1-z+z^2-tz)G+(1-z)(1-tz) = 0 (for the explicit form of G see the Maple program).
The trivariate g.f. g=g(x,y,z) of Dyck paths with respect to number of peak plateaux, number of peaks in the peak plateaux and semilength, marked, by x, y and z, respectively satisfies g=1+zg[g+xyz/(1-yz)-z/(1-z)].
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