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 A288686 Number of Dyck paths of semilength n such that no positive level has fewer than ten peaks. 2
 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 618, 11695, 112434, 665219, 2756389, 8890492, 24410518, 60972735, 144203914, 329766287, 737405644, 1623087349, 3531560786, 7633789153, 16745585892, 41482511559, 152244106469, 886899776271 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,22 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, 10, j], {j, 10, 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 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, 10, j) for j in range(10, n + 1)) print([a(n) for n in range(41)]) # Indranil Ghosh, Aug 10 2017 CROSSREFS Column k=10 of A288386. Cf. A000108. Sequence in context: A218999 A220368 A220312 * A013587 A126159 A220992 Adjacent sequences:  A288683 A288684 A288685 * A288687 A288688 A288689 KEYWORD nonn AUTHOR Alois P. Heinz, Jun 13 2017 STATUS approved

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Last modified May 26 22:58 EDT 2020. Contains 334634 sequences. (Running on oeis4.)