A112307 Triangle read by rows: T(n,k) is number of Dyck paths of semilength n with height of second peak equal to k (n>=1; 0<=k<=n-1).

1, 1, 1, 1, 2, 2, 1, 4, 6, 3, 1, 9, 16, 12, 4, 1, 23, 44, 39, 20, 5, 1, 65, 128, 123, 76, 30, 6, 1, 197, 392, 393, 268, 130, 42, 7, 1, 626, 1250, 1284, 928, 505, 204, 56, 8, 1, 2056, 4110, 4287, 3216, 1880, 864, 301, 72, 9, 1, 6918, 13834, 14583, 11224, 6885, 3438, 1379

Offset

1,5

Comments

Row sums are the Catalan numbers (A000108). T(n,0)=1 (paths have only one peak); The g.f. for column k is kz^(k+1)*c^k/(1-z), where c=[1-sqrt(1-4z)]/(2z) is the Catalan function. T(n,1)=A014137(n-1); T(n,2)=2*A014138(n-3); T(n,3)=3*A001453(n-2); T(n,4)=4*A114277(n-5); Sum(k*T(n,k), k=0..n-1)=A112308(n-2).

Links

Table of n, a(n) for n=1..63.

Formula

G.f.=[(1-tzc)^2+tz^2*c]/[(1-z)(1-tzc)^2]-1, where c=[1-sqrt(1-4z)]/(2z) is the Catalan function.

Example

T(4,1)=4 because we have UDUDUDUD, UDUDUUDD, UUDDUDUD and UUUDDDUD, where U=(1,1), D=(1,-1).
Triangle begins:
1;
1,1;
1,2,2;
1,4,6,3;
1,9,16,12,4;

Maple

G:=((1-t*z*c)^2+t*z^2*c)/(1-z)/(1-t*z*c)^2-1: c:=(1-sqrt(1-4*z))/2/z: Gser:=simplify(series(G, z=0, 15)): for n from 1 to 12 do P[n]:=coeff(Gser, z^n) od: for n from 1 to 12 do seq(coeff(t*P[n], t^j), j=1..n) od; # yields sequence in triangular form

Crossrefs

Cf. A000108, A014137, A014138, A001453, A114277, A112308.

Sequence in context: A119468 A175136 A091869 * A228336 A111062 A369632

Adjacent sequences: A112304 A112305 A112306 * A112308 A112309 A112310

Keyword

nonn,tabl

Author

Emeric Deutsch, Nov 30 2005

Status

approved