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A288318 Number T(n,k) of Dyck paths of semilength n such that each positive level has exactly k peaks; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A288318 #22 Oct 18 2018 15:44:38
%S A288318 1,0,1,0,0,1,0,2,0,1,0,0,0,0,1,0,4,3,0,0,1,0,6,6,0,0,0,1,0,8,0,4,0,0,
%T A288318 0,1,0,24,9,20,0,0,0,0,1,0,52,54,20,5,0,0,0,0,1,0,96,138,0,45,0,0,0,0,
%U A288318 0,1,0,212,207,16,105,6,0,0,0,0,0,1,0,504,360,200,70,84,0,0,0,0,0,0,1
%N A288318 Number T(n,k) of Dyck paths of semilength n such that each positive level has exactly k peaks; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A288318 T(n,k) is defined for all n,k >= 0.  The triangle contains only the terms for k<=n. T(0,k) = 1 and T(n,k) = 0 if k > n > 0.
%H A288318 Alois P. Heinz, <a href="/A288318/b288318.txt">Rows n = 0..140, flattened</a>
%H A288318 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A288318 T(n,n) = 1.
%F A288318 T(n+1,n) = 0.
%F A288318 T(2*n+1,n) = (n+1) for n>0.
%F A288318 T(2*n+2,n) = A005564(n+1) for n>1.
%F A288318 T(3*n,n) = A000984(n) = binomial(2*n,n).
%F A288318 T(3*n+1,n) = 0.
%F A288318 T(3*n+2,n) = (n+1)^2 for n>0.
%e A288318 . T(5,1) = 4:
%e A288318 .               /\        /\          /\        /\
%e A288318 .            /\/  \      /  \/\    /\/  \      /  \/\
%e A288318 .         /\/      \  /\/      \  /      \/\  /      \/\ .
%e A288318 .
%e A288318 . T(5,2) = 3:
%e A288318 .              /\/\      /\/\      /\/\
%e A288318 .         /\/\/    \  /\/    \/\  /    \/\/\  .
%e A288318 .
%e A288318 Triangle T(n,k) begins:
%e A288318   1;
%e A288318   0,  1;
%e A288318   0,  0,  1;
%e A288318   0,  2,  0,  1;
%e A288318   0,  0,  0,  0, 1;
%e A288318   0,  4,  3,  0, 0, 1;
%e A288318   0,  6,  6,  0, 0, 0, 1;
%e A288318   0,  8,  0,  4, 0, 0, 0, 1;
%e A288318   0, 24,  9, 20, 0, 0, 0, 0, 1;
%e A288318   0, 52, 54, 20, 5, 0, 0, 0, 0, 1;
%p A288318 b:= proc(n, k, j) option remember;
%p A288318      `if`(n=j, 1, add(b(n-j, k, i)*(binomial(i, k)
%p A288318       *binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
%p A288318     end:
%p A288318 T:= (n, k)-> `if`(n=0, 1, b(n, k$2)):
%p A288318 seq(seq(T(n, k), k=0..n), n=0..14);
%t A288318 b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[b[n - j, k, i]*(Binomial[i, k]*Binomial[j - 1, i - 1 - k]), {i, 1, Min[j + k, n - j]}]];
%t A288318 T[n_, k_] := If[n == 0, 1, b[n, k, k]];
%t A288318 Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 25 2018, translated from Maple *)
%Y A288318 Columns k=0-10 give: A000007, A287846, A287845, A288319, A288320, A288321, A288322, A288323, A288324, A288325, A288326.
%Y A288318 Row sums give A287987.
%Y A288318 Cf. A000108, A000984, A005564, A288108, A288940.
%K A288318 nonn,tabl
%O A288318 0,8
%A A288318 _Alois P. Heinz_, Jun 07 2017