The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A288387 Number T(n,k) of Dyck paths of semilength n such that the minimal number of peaks over all positive levels equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 13
 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 8, 5, 0, 0, 1, 25, 13, 3, 0, 0, 1, 83, 35, 13, 0, 0, 0, 1, 282, 112, 30, 4, 0, 0, 0, 1, 971, 368, 61, 29, 0, 0, 0, 0, 1, 3386, 1208, 172, 90, 5, 0, 0, 0, 0, 1, 11940, 3992, 619, 188, 56, 0, 0, 0, 0, 0, 1, 42504, 13449, 2241, 345, 240, 6, 0, 0, 0, 0, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS 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. T(0,0) = 1 by convention. LINKS Alois P. Heinz, Rows n = 0..140, flattened Wikipedia, Counting lattice paths FORMULA T(0,0) = 1, T(n,k) = A288386(n,k) - A288386(n,k+1). T(2n,n-1) = A218152(n) for n>1. T(2n,n) = A000007(n). T(2n+1,n) = A000027(n+1) for n>0. EXAMPLE . T(4,1) = 5: . /\ /\ /\/\ /\ /\/\ . /\/\/ \ /\/ \/\ /\/ \ / \/\/\ / \/\ . . Triangle T(n,k) begins: : 1; : 0, 1; : 1, 0, 1; : 2, 2, 0, 1; : 8, 5, 0, 0, 1; : 25, 13, 3, 0, 0, 1; : 83, 35, 13, 0, 0, 0, 1; : 282, 112, 30, 4, 0, 0, 0, 1; : 971, 368, 61, 29, 0, 0, 0, 0, 1; : 3386, 1208, 172, 90, 5, 0, 0, 0, 0, 1; MAPLE b:= proc(n, k, j) option remember; `if`(j=n, 1, add(add(binomial(i, m)*binomial(j-1, i-1-m), m=max(k, i-j)..i-1)*b(n-j, k, i), i=1..n-j)) end: A:= proc(n, k) option remember; `if`(n=0, 1, add(b(n, k, j), j=k..n)) end: T:= (n, k)-> `if`(n=k, 1, A(n, k)-A(n, k+1)): seq(seq(T(n, k), k=0..n), n=0..14); MATHEMATICA b[n_, k_, j_] := b[n, k, j] = If[j==n, 1, Sum[Sum[Binomial[i, m]*Binomial[ j-1, i-1-m], {m, Max[k, i - j], i - 1}]*b[n - j, k, i], {i, 1, n - j}]]; A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[b[n, k, j], {j, k, n}]]; T[n_, k_] := If[n == k, 1, A[n, k] - A[n, k + 1]]; 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: A288539, A288540, A288541, A288542, A288543, A288544, A288545, A288546, A288547, A288548, A288549. Row sums give A000108. Main diagonal and first lower diagonal give: A000012, A000004. Cf. A000007, A000027, A218152, A287822, A288386. Sequence in context: A065600 A029583 A011289 * A225678 A141720 A353449 Adjacent sequences: A288384 A288385 A288386 * A288388 A288389 A288390 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jun 08 2017 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 2 23:30 EDT 2023. Contains 365841 sequences. (Running on oeis4.)