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 A288684 Number of Dyck paths of semilength n such that no positive level has fewer than eight peaks. 2
 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 335, 4241, 27915, 117971, 373845, 1002089, 2456082, 5725439, 12935530, 28622833, 62588817, 139046970, 353173119, 1305216091, 7035422989, 41539474198, 227550374938, 1115122502718, 4917988882292 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,18 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, 8, j], {j, 8, 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, 8, j) for j in xrange(8, n + 1)]) print map(a, xrange(41)) # Indranil Ghosh, Aug 10 2017 CROSSREFS Column k=8 of A288386. Cf. A000108. Sequence in context: A257132 A218996 A113082 * A046747 A006426 A029698 Adjacent sequences:  A288681 A288682 A288683 * A288685 A288686 A288687 KEYWORD nonn AUTHOR Alois P. Heinz, Jun 13 2017 STATUS approved

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Last modified January 23 03:08 EST 2019. Contains 319370 sequences. (Running on oeis4.)