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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. 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 (list; table; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A105292 A273342 A276067 * A125178 A101975 A136388
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

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Last modified March 29 08:13 EDT 2024. Contains 371265 sequences. (Running on oeis4.)