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 A288108 Number T(n,k) of Dyck paths of semilength n such that each level has exactly k peaks or no peaks; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 13
 1, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 5, 2, 1, 1, 0, 13, 5, 3, 1, 1, 0, 31, 15, 4, 4, 1, 1, 0, 71, 27, 10, 7, 5, 1, 1, 0, 181, 76, 36, 11, 11, 6, 1, 1, 0, 447, 196, 83, 22, 19, 16, 7, 1, 1, 0, 1111, 548, 225, 81, 32, 31, 22, 8, 1, 1, 0, 2799, 1388, 573, 235, 60, 56, 48, 29, 9, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS 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. LINKS Alois P. Heinz, Rows n = 0..140, flattened Wikipedia, Counting lattice paths EXAMPLE . T(5,2) = 5:                                        /\/\ .                                       /\  /\      /    \ .      /\/\      /\/\      /\/\        /  \/  \    /      \ . /\/\/    \  /\/    \/\  /    \/\/\  /        \  /        \ . . . T(5,3) = 3: .                                       /\/\/\ .              /\  /\/\    /\/\  /\    /      \ .             /  \/    \  /    \/  \  /        \ . . Triangle T(n,k) begins:   1;   0,   1;   0,   1,  1;   0,   3,  1,  1;   0,   5,  2,  1,  1;   0,  13,  5,  3,  1,  1;   0,  31, 15,  4,  4,  1, 1;   0,  71, 27, 10,  7,  5, 1, 1;   0, 181, 76, 36, 11, 11, 6, 1, 1; MAPLE b:= proc(n, k, j) option remember; `if`(n=j, 1, add(       b(n-j, k, i)*(binomial(j-1, i-1)+binomial(i, k)        *binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))     end: T:= (n, k)-> b(n, k\$2): seq(seq(T(n, k), k=0..n), n=0..14); MATHEMATICA b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[b[n - j, k, i]*(Binomial[j - 1, i - 1] + Binomial[i, k]*Binomial[j - 1, i - 1 - k]), {i, 1, Min[j + k, n - j]}]]; T[n_, k_] := b[n, k, k]; Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 25 2018, translated from Maple *) CROSSREFS Columns k=0-10 give: A000007, A281874, A287843, A288110, A288111, A288112, A288113, A288114, A288115, A288116, A288117. Row sums give A288109. T(2n,n) gives A156043. Cf. A000108, A288318. Sequence in context: A130115 A191582 A130160 * A287822 A162169 A216954 Adjacent sequences:  A288105 A288106 A288107 * A288109 A288110 A288111 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jun 05 2017 STATUS approved

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Last modified August 8 19:29 EDT 2020. Contains 336298 sequences. (Running on oeis4.)