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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 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, 10, j) for j in xrange(10, n + 1)])

print map(a, xrange(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 January 20 16:44 EST 2019. Contains 319335 sequences. (Running on oeis4.)