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A371928
T(n,k) is the total number of levels in all Dyck paths of semilength n containing exactly k path nodes; triangle T(n,k), n>=0, 1<=k<=n+1, read by rows.
5
1, 1, 1, 1, 3, 1, 3, 5, 6, 1, 8, 15, 13, 11, 1, 23, 44, 43, 29, 20, 1, 71, 134, 138, 106, 62, 37, 1, 229, 427, 446, 371, 248, 132, 70, 1, 759, 1408, 1478, 1275, 941, 571, 283, 135, 1, 2566, 4753, 5017, 4410, 3437, 2331, 1310, 611, 264, 1, 8817, 16321, 17339, 15458, 12426, 9027, 5709, 3002, 1324, 521, 1
OFFSET
0,5
COMMENTS
A Dyck path of semilength n has 2n+1 = A005408(n) nodes.
LINKS
FORMULA
Sum_{k=1..n+1} k * T(n,k) = A001700(n) = A005408(n) * A000108(n).
EXAMPLE
In the A000108(3) = 5 Dyck paths of semilength 3 there are 3 levels with 1 node, 5 levels with 2 nodes, 6 levels with 3 nodes, and 1 level with 4 nodes.
1
2 /\ 2 1 1
2 / \ 3 /\/\ 3 /\ 3 /\ 3
2 / \ 2 / \ 3 / \/\ 3 /\/ \ 4 /\/\/\ .
So row 3 is [3, 5, 6, 1].
Triangle T(n,k) begins:
1;
1, 1;
1, 3, 1;
3, 5, 6, 1;
8, 15, 13, 11, 1;
23, 44, 43, 29, 20, 1;
71, 134, 138, 106, 62, 37, 1;
229, 427, 446, 371, 248, 132, 70, 1;
759, 1408, 1478, 1275, 941, 571, 283, 135, 1;
2566, 4753, 5017, 4410, 3437, 2331, 1310, 611, 264, 1;
8817, 16321, 17339, 15458, 12426, 9027, 5709, 3002, 1324, 521, 1;
...
MAPLE
g:= proc(x, y, p) (h-> `if`(x=0, add(z^coeff(h, z, i)
, i=0..degree(h)), b(x, y, h)))(p+z^y) end:
b:= proc(x, y, p) option remember; `if`(y+2<=x,
g(x-1, y+1, p), 0)+`if`(y>0, g(x-1, y-1, p), 0)
end:
T:= n-> (p-> seq(coeff(p, z, i), i=1..n+1))(g(2*n, 0$2)):
seq(T(n), n=0..10);
CROSSREFS
Columns k=1-2 give: A152880, A371903.
Row sums give A261003.
T(n+1,n+1) gives A006127.
Sequence in context: A345943 A342342 A182600 * A179760 A334852 A160552
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Apr 14 2024
STATUS
approved