A125177 Triangle read by rows: T(n,0)=C(2n,n)/(n+1) for n>=0; T(n,n+1)=0; T(n,k)=T(n-1,k)+T(n-1,k-1) for 1<=k<=n.

1, 1, 1, 2, 2, 1, 5, 4, 3, 1, 14, 9, 7, 4, 1, 42, 23, 16, 11, 5, 1, 132, 65, 39, 27, 16, 6, 1, 429, 197, 104, 66, 43, 22, 7, 1, 1430, 626, 301, 170, 109, 65, 29, 8, 1, 4862, 2056, 927, 471, 279, 174, 94, 37, 9, 1, 16796, 6918, 2983, 1398, 750, 453, 268, 131, 46, 10, 1, 58786

Offset

0,4

Comments

Column k (k>=1) starts with 0, followed by the partial sums of column k-1. Row sums yield A126221.

Indexing n and k from 1 instead of from 0, T(n,k) is the number of Dyck n-paths whose first peak is at height k and whose first component avoids DUU. A primitive Dyck path is one whose only return (to ground level) is at the end. The interior returns of a general Dyck path split the path into a list of primitive Dyck paths, called its components. For example, UUDDUD has components UUDD, UD and T(4,2) = 4 counts UUDUDUDD, UUDDUUDD, UUDDUDUD, UUDUDDUD (but not UUDUUDDD because its first component contains a DUU). - David Callan, Jan 17 2007

Riordan array (c(x),x/(1-x)), c(x) the g.f. of A000108. Equal to ((1-x)*c(x),x)*A007318. [Paul Barry, May 06 2009]

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Formula

G.f.: G(t,x)=(1-x)[1-sqrt(1-4x)]/[2x(1-x-tx)].

T(n,k) = Sum_{j=0..n} C(n-j,k)*if(j=0,0^j, A000108(j)-A000108(j-1)). [Paul Barry, May 06 2009]

T(n,k) = Sum_{i=0..n-k} binomial(n-i-1,n-k-i)*A000108(i). - Vladimir Kruchinin, Nov 03 2016

Example

First few rows of the triangle are:
  1;
  1, 1;
  2, 2, 1;
  5, 4, 3, 1;
  14, 9, 7, 4, 1;
  42, 23, 16, 11, 5, 1;
  ...
(5,3) = 16 = 7 + 9 = (4,3) + (4,2).
From Paul Barry, May 06 2009: (Start)
Production matrix is
  1, 1,
  1, 1, 1,
  1, 0, 1, 1,
  2, 0, 0, 1, 1,
  4, 0, 0, 0, 1, 1,
  9, 0, 0, 0, 0, 1, 1,
  21, 0, 0, 0, 0, 0, 1, 1,
  51, 0, 0, 0, 0, 0, 0, 1, 1,
  127, 0, 0, 0, 0, 0, 0, 0, 1, 1 (End)

Maple

T:=proc(n, k) if k=0 then binomial(2*n, n)/(n+1) elif n=0 then 0 else T(n-1, k)+T(n-1, k-1) fi end: for n from 0 to 11 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
G:=(1-x)*(1-sqrt(1-4*x))/2/x/(1-x-t*x): Gser:=simplify(series(G, x=0, 15)): for n from 0 to 11 do P[n]:=sort(coeff(Gser, x, n)) od: for n from 0 to 11 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form

Prog

(Maxima) T(n, k)=sum((binomial(2*i, i)*binomial(n-i-1, n-k-i))/(i+1), i, 0, n-k); /* Vladimir Kruchinin, Nov 03 2016 */

Crossrefs

Cf. A000108, A125178, A126221.

Sequence in context: A105292 A273342 A276067 * A125178 A101975 A136388

Adjacent sequences: A125174 A125175 A125176 * A125178 A125179 A125180

Keyword

nonn,tabl

Author

Gary W. Adamson, Nov 22 2006

Extensions

Edited by Emeric Deutsch, Dec 28 2006

Definition amended by Georg Fischer, Jun 16 2022

Status

approved