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 A288683 Number of Dyck paths of semilength n such that no positive level has fewer than seven peaks. 2
 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 9, 234, 2350, 12567, 44971, 127475, 320491, 756677, 1720610, 3821223, 8436508, 19793620, 59810128, 268048977, 1458971589, 7720465569, 36927931597, 159094351283, 626621217546, 2296016964863, 7949275945740 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,16 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, 7, j], {j, 7, n}]]; Table[a[n], {n, 0, 35}] (* 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, 7, j) for j in range(7, n + 1)]) print([a(n) for n in range(36)]) # Indranil Ghosh, Aug 10 2017 CROSSREFS Column k=7 of A288386. Cf. A000108. Sequence in context: A165389 A153223 A276536 * A302914 A157569 A279492 Adjacent sequences: A288680 A288681 A288682 * A288684 A288685 A288686 KEYWORD nonn AUTHOR Alois P. Heinz, Jun 13 2017 STATUS approved

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Last modified May 19 09:42 EDT 2024. Contains 372683 sequences. (Running on oeis4.)