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 A288682 Number of Dyck paths of semilength n such that no positive level has fewer than six peaks. 2
 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 8, 156, 1213, 5232, 16091, 41834, 100320, 229851, 513699, 1166304, 3068322, 11294356, 54431307, 271824026, 1253186445, 5233138157, 20031588131, 71538367677, 242280234545, 789260222205, 2507719402158, 7900354628357 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,14 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..300 Wikipedia, Counting lattice paths 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, 6, j], {j, 6, n}]]; Table[a[n], {n, 0, 35}] (* Indranil Ghosh, Aug 10 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 xrange(max(k, i - j), i)])*b(n - j, k, i) for i in xrange(1, n - j + 1)]) def a(n): return 1 if n==0 else sum([b(n, 6, j) for j in xrange(6, n + 1)]) print map(a, xrange(36)) # Indranil Ghosh, Aug 10 2017 CROSSREFS Column k=6 of A288386. Cf. A000108. Sequence in context: A302959 A188408 A089669 * A268543 A113668 A120348 Adjacent sequences:  A288679 A288680 A288681 * A288683 A288684 A288685 KEYWORD nonn AUTHOR Alois P. Heinz, Jun 13 2017 STATUS approved

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Last modified October 18 04:46 EDT 2019. Contains 328145 sequences. (Running on oeis4.)