cp's OEIS Frontend

A287822 Number T(n,k) of Dyck paths of semilength n such that the maximal number of peaks per level equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A287822 #22 Mar 14 2024 19:01:47
%S A287822 1,0,1,0,1,1,0,3,1,1,0,5,7,1,1,0,13,18,9,1,1,0,31,59,29,11,1,1,0,71,
%T A287822 193,112,38,13,1,1,0,181,616,405,163,48,15,1,1,0,447,1955,1514,648,
%U A287822 220,59,17,1,1,0,1111,6244,5565,2571,925,288,71,19,1,1
%N A287822 Number T(n,k) of Dyck paths of semilength n such that the maximal number of peaks per level equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A287822 T(n,k) is defined for all n,k >= 0.  The triangle contains only the terms for k<=n. T(n,k) = 0 if k>n.
%H A287822 Alois P. Heinz, <a href="/A287822/b287822.txt">Rows n = 0..100, flattened</a>
%H A287822 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A287822 T(n,k) = A287847(n,k) - A287847(n,k-1) for k>0, T(n,0) = A000007(n).
%e A287822 . T(4,1) = 5:                                             /\
%e A287822 .                  /\        /\      /\        /\        /  \
%e A287822 .                 /  \    /\/  \    /  \      /  \/\    /    \
%e A287822 .              /\/    \  /      \  /    \/\  /      \  /      \ .
%e A287822 .
%e A287822 . T(4,2) = 7:       /\      /\        /\/\    /\        /\  /\
%e A287822 .              /\/\/  \  /\/  \/\  /\/    \  /  \/\/\  /  \/  \ .
%e A287822 .
%e A287822 .                          /\/\
%e A287822 .               /\/\      /    \
%e A287822 .              /    \/\  /      \  .
%e A287822 .
%e A287822 . T(4,3) = 1:   /\/\/\
%e A287822 .              /      \  .
%e A287822 .
%e A287822 . T(4,4) = 1:  /\/\/\/\  .
%e A287822 .
%e A287822 Triangle T(n,k) begins:
%e A287822   1;
%e A287822   0,   1;
%e A287822   0,   1,    1;
%e A287822   0,   3,    1,    1;
%e A287822   0,   5,    7,    1,   1;
%e A287822   0,  13,   18,    9,   1,   1;
%e A287822   0,  31,   59,   29,  11,   1,  1;
%e A287822   0,  71,  193,  112,  38,  13,  1,  1;
%e A287822   0, 181,  616,  405, 163,  48, 15,  1, 1;
%e A287822   0, 447, 1955, 1514, 648, 220, 59, 17, 1, 1;
%e A287822   ...
%p A287822 b:= proc(n, k, j) option remember; `if`(j=n, 1, add(
%p A287822       b(n-j, k, i)*add(binomial(i, m)*binomial(j-1, i-1-m),
%p A287822        m=max(0, i-j)..min(k, i-1)), i=1..min(j+k, n-j)))
%p A287822     end:
%p A287822 A:= proc(n, k) option remember; `if`(n=0, 1, (m->
%p A287822       add(b(n, m, j), j=1..m))(min(n, k)))
%p A287822     end:
%p A287822 T:= (n, k)-> A(n, k)- `if`(k=0, 0, A(n, k-1)):
%p A287822 seq(seq(T(n, k), k=0..n), n=0..12);
%t A287822 b[n_, k_, j_] := b[n, k, j] = If[j == n, 1, Sum[b[n - j, k, i]*Sum[ Binomial[i, m]*Binomial[j - 1, i - 1 - m], {m, Max[0, i - j], Min[k, i - 1]}], {i, 1, Min[j + k, n - j]}]];
%t A287822 A[n_, k_] := A[n, k] = If[n==0, 1, Sum[b[n, #, j], {j, 1, #}]&[Min[n, k]]];
%t A287822 T[n_, k_] := A[n, k] - If[k==0, 0, A[n, k - 1]];
%t A287822 Table[T[n, k], {n, 0, 12}, { k, 0, n}] // Flatten (* _Jean-François Alcover_, May 25 2018, translated from Maple *)
%Y A287822 Columns k=0-10 give: A000007, A281874 (for n>0), A288743, A288744, A288745, A288746, A288747, A288748, A288749, A288750, A288751.
%Y A287822 Row sums give A000108.
%Y A287822 T(2n,n) gives A287860.
%Y A287822 Cf. A287847.
%K A287822 nonn,tabl
%O A287822 0,8
%A A287822 _Alois P. Heinz_, Jun 01 2017