A125178 -id:A125178 - cp's OEIS frontend

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

Gary W. Adamson, Nov 22 2006

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]

			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)

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

Cf. A000108, A125178, A126221.

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

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 */

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

Edited by Emeric Deutsch, Dec 28 2006

Definition amended by Georg Fischer, Jun 16 2022

cp's OEIS Frontend

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.

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