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 A288685 Number of Dyck paths of semilength n such that no positive level has fewer than nine peaks. 2
 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 462, 7217, 57783, 289400, 1043781, 3042593, 7833174, 18821247, 43417043, 97550980, 215243289, 469069428, 1020806036, 2342090587, 6886047798, 32238887181, 199504672863, 1232775909721, 6881782444707 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,20 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, 9, j], {j, 9, n}]]; Table[a[n], {n, 0, 40}] (* 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, 9, j) for j in xrange(9, n + 1)]) print map(a, xrange(41)) # Indranil Ghosh, Aug 10 2017 CROSSREFS Column k=9 of A288386. Cf. A000108. Sequence in context: A175158 A180087 A233219 * A068235 A247598 A303207 Adjacent sequences:  A288682 A288683 A288684 * A288686 A288687 A288688 KEYWORD nonn AUTHOR Alois P. Heinz, Jun 13 2017 STATUS approved

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Last modified January 19 15:53 EST 2019. Contains 319307 sequences. (Running on oeis4.)