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 A287901 Number of Dyck paths of semilength n such that each positive level up to the highest nonempty level has at least one peak. 4
 1, 1, 1, 3, 6, 17, 49, 147, 459, 1476, 4856, 16282, 55466, 191474, 668510, 2356944, 8380944, 30025814, 108289093, 392871484, 1432934360, 5251507624, 19329771911, 71430479820, 264914270527, 985737417231, 3679051573264, 13769781928768, 51670641652576 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..300 Wikipedia, Counting lattice paths EXAMPLE . a(3) = 3: . /\ /\ . /\/\/\ /\/ \ / \/\ . . . a(4) = 6: . /\ /\ /\/\ /\ /\/\ . /\/\/\/\ /\/\/ \ /\/ \/\ /\/ \ / \/\/\ / \/\ . 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, n - j}]]; a[n_]:=If[n==0, 1, Sum[b[n, 1, j], {j, n}]]; Table[a[n], {n, 0, 30}] (* Indranil Ghosh, Aug 09 2017 *) PROG (Python) from sympy.core.cache import cacheit from sympy import binomial @cacheit def b(n, k, j): return 1 if j==n else sum([sum([binomial(i, m)*binomial(j - 1, i - 1 - m) for m in range(max(k, i - j), i)])*b(n - j, k, i) for i in range(1, n - j + 1)]) def a(n): return 1 if n==0 else sum([b(n, 1, j) for j in range(1, n + 1)]) print([a(n) for n in range(31)]) # Indranil Ghosh, Aug 09 2017 CROSSREFS Column k=1 of A288386. Cf. A000108, A281874, A287846. Sequence in context: A204517 A307685 A360273 * A354878 A143093 A117712 Adjacent sequences: A287898 A287899 A287900 * A287902 A287903 A287904 KEYWORD nonn AUTHOR Alois P. Heinz, Jun 02 2017 STATUS approved

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Last modified March 30 11:15 EDT 2023. Contains 361618 sequences. (Running on oeis4.)