

A125177


Triangle read by rows: T(n,0)=C(2n,n)/(n+1) for n>=0; T(0,k)=0 for k>=1; T(n,k)=T(n1,k)+T(n1,k1) for n>=1, k>=1.


4



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
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OFFSET

0,4


COMMENTS

Column k (k>=1) starts with 0, followed by the partial sums of column k1. Row sums yield A126221.
Indexing n and k from 1 instead of from 0, T(n,k) is the number of Dyck npaths 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/(1x)), c(x) the g.f. of A000108. Equal to ((1x)*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)=(1x)[1sqrt(14x)]/[2x(1xtx)].
T(n,k) = Sum_{j=0..n} C(nj,k)*if(j=0,0^j, A000108(j)A000108(j1)). [Paul Barry, May 06 2009]
T(n,k) = Sum_{i=0..nk} binomial(ni1,nki)*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).
Contribution 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(n1, k)+T(n1, k1) fi end: for n from 0 to 11 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
G:=(1x)*(1sqrt(14*x))/2/x/(1xt*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

(Maaxima) T(n, k)=sum((binomial(2*i, i)*binomial(ni1, nki))/(i+1), i, 0, nk); /* 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


STATUS

approved



