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A287822
Number T(n,k) of Dyck paths of semilength n such that the maximal number of peaks per level equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
13
1, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 5, 7, 1, 1, 0, 13, 18, 9, 1, 1, 0, 31, 59, 29, 11, 1, 1, 0, 71, 193, 112, 38, 13, 1, 1, 0, 181, 616, 405, 163, 48, 15, 1, 1, 0, 447, 1955, 1514, 648, 220, 59, 17, 1, 1, 0, 1111, 6244, 5565, 2571, 925, 288, 71, 19, 1, 1
OFFSET
0,8
COMMENTS
T(n,k) is defined for all n,k >= 0. The triangle contains only the terms for k<=n. T(n,k) = 0 if k>n.
LINKS
FORMULA
T(n,k) = A287847(n,k) - A287847(n,k-1) for k>0, T(n,0) = A000007(n).
EXAMPLE
. T(4,1) = 5: /\
. /\ /\ /\ /\ / \
. / \ /\/ \ / \ / \/\ / \
. /\/ \ / \ / \/\ / \ / \ .
.
. T(4,2) = 7: /\ /\ /\/\ /\ /\ /\
. /\/\/ \ /\/ \/\ /\/ \ / \/\/\ / \/ \ .
.
. /\/\
. /\/\ / \
. / \/\ / \ .
.
. T(4,3) = 1: /\/\/\
. / \ .
.
. T(4,4) = 1: /\/\/\/\ .
.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 3, 1, 1;
0, 5, 7, 1, 1;
0, 13, 18, 9, 1, 1;
0, 31, 59, 29, 11, 1, 1;
0, 71, 193, 112, 38, 13, 1, 1;
0, 181, 616, 405, 163, 48, 15, 1, 1;
0, 447, 1955, 1514, 648, 220, 59, 17, 1, 1;
...
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:
A:= proc(n, k) option remember; `if`(n=0, 1, (m->
add(b(n, m, j), j=1..m))(min(n, k)))
end:
T:= (n, k)-> A(n, k)- `if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..12);
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, 1, Min[j + k, n - j]}]];
A[n_, k_] := A[n, k] = If[n==0, 1, Sum[b[n, #, j], {j, 1, #}]&[Min[n, k]]];
T[n_, k_] := A[n, k] - If[k==0, 0, A[n, k - 1]];
Table[T[n, k], {n, 0, 12}, { k, 0, n}] // Flatten (* Jean-François Alcover, May 25 2018, translated from Maple *)
CROSSREFS
Columns k=0-10 give: A000007, A281874 (for n>0), A288743, A288744, A288745, A288746, A288747, A288748, A288749, A288750, A288751.
Row sums give A000108.
T(2n,n) gives A287860.
Cf. A287847.
Sequence in context: A363756 A130160 A288108 * A162169 A216954 A124801
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jun 01 2017
STATUS
approved