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 A288749 Number of Dyck paths of semilength n such that the maximal number of peaks per level equals eight. 2
 1, 1, 19, 84, 461, 2222, 10577, 48943, 222627, 997735, 4417674, 19359659, 84099436, 362570722, 1552681071, 6609823112, 27989970166, 117967914457, 495087382572, 2069827499508, 8623283249034, 35811917284318, 148289870077879, 612382134256433, 2522591250558641 (list; graph; refs; listen; history; text; internal format)
 OFFSET 8,3 LINKS Alois P. Heinz, Table of n, a(n) for n = 8..1000 Wikipedia, Counting lattice paths MAPLE b:= proc(n, k, j) option remember; `if`(j=n, 1, add(       b(n-j, k, i)*add(binomial(i, m)*binomial(j-1, i-1-m),        m=max(0, i-j)..min(k, i-1)), i=1..min(j+k, n-j)))     end: g:= proc(n, k) option remember; add(b(n, k, j), j=1..k) end: a:= n-> g(n, 8)-g(n, 7): seq(a(n), n=8..35); MATHEMATICA b[n_, k_, j_]:=b[n, k, j]=If[j==n, 1, Sum[b[n - j, k, i] Sum[Binomial[i, m] Binomial[j - 1, i - 1 - m], {m, Max[0, i - j], Min[k, i - 1]}], {i, Min[j + k, n - j]}]]; g[n_, k_]:=Sum[b[n, k, j], {j, k}]; Table[g[n, 8] - g[n, 7], {n, 8, 35}] (* Indranil Ghosh, Aug 08 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([b(n - j, k, i)*sum([binomial(i, m)*binomial(j - 1, i - 1 - m) for m in xrange(max(0, i - j), min(k, i - 1) + 1)]) for i in xrange(1, min(j + k, n - j) + 1)]) def g(n, k): return sum([b(n, k, j) for j in xrange(1, k + 1)]) def a(n): return g(n, 8) - g(n, 7) print map(a, xrange(8, 36)) # Indranil Ghosh, Aug 08 2017 CROSSREFS Column k=8 of A287822. Cf. A000108. Sequence in context: A036564 A062639 A209369 * A039609 A063496 A027848 Adjacent sequences:  A288746 A288747 A288748 * A288750 A288751 A288752 KEYWORD nonn AUTHOR Alois P. Heinz, Jun 14 2017 STATUS approved

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Last modified January 17 10:08 EST 2019. Contains 319218 sequences. (Running on oeis4.)