A114516 Triangle read by rows: T(n,k) is number of Dyck paths of semilength n in which the number of ascents and descents of length 1 is equal to k (0<=k<=2n).

1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 3, 0, 0, 0, 1, 2, 2, 3, 0, 6, 0, 0, 0, 1, 4, 4, 9, 8, 6, 0, 10, 0, 0, 0, 1, 8, 12, 24, 16, 26, 20, 10, 0, 15, 0, 0, 0, 1, 17, 32, 58, 64, 81, 40, 60, 40, 15, 0, 21, 0, 0, 0, 1, 37, 82, 159, 196, 221, 210, 205, 80, 120, 70, 21, 0, 28, 0, 0, 0, 1, 82, 212, 428, 576

Offset

0,12

Comments

Row n has 2n+1 terms.

Row sums are the Catalan numbers (A000108).

T(n,0)=A004148(n-1) for n>=2 (the DNA secondary structure numbers).

Sum(k*T(n,k),k=0..2n+1)=2*binomial(2n-2,n-1) (2*A000984).

The trivariate g.f. g=g(t,s,z) of the Dyck paths, where z marks semilength and t(s) marks number of ascents (descents) of length 1, satisfies z(1+tz-tsz)(1+sz-tsz)g^2 - [1+(1-ts)z-(1-t)(1-s)z^2]g+1=0. Clearly, equation for G is obtained from here by taking s=t.

Links

Table of n, a(n) for n=0..84.

Formula

G.f.: G=G(t, z) satisfies z*(1+t*z-z*t^2*z)^2*G^2-(1+z-z^2-t^2*z+2*t*z^2-t^2*z^2)*G+ 1=0.

Example

T(5,3)=8 because we have UU(DUD)UUDDD, (UD)UU(D)UUDDD, UU(D)UUDDD(UD),
UUU(DU)DD(U)DD and their reflections; here U=(1,1) and D=(1,-1).
Triangle begins:
1;
0,0,1;
1,0,0,0,1;
1,0,3,0,0,0,1;
2,2,3,0,6,0,0,0,1;
4,4,9,8,6,0,10,0,0,0,1;

Maple

G:=1/2/(z^3*t^4+z^3*t^2-2*z^2*t^2+2*z^2*t+z-2*z^3*t^3)*(-z^2*t^2+z-z^2-z*t^2+2*z^2*t+1-sqrt(1+z^4*t^4+6*z^4*t^2-4*z^4*t^3+4*z^3*t-4*z^4*t+z^4-2*z^3*t^4-2*z+4*z^3*t^3-4*z^2*t+z^2*t^4-2*z^3+4*z^2*t^2-2*z*t^2-z^2-4*z^3*t^2)): Gser:=simplify(series(G, z=0, 13)): P[0]:=1: for n from 1 to 9 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 9 do seq(coeff(t*P[n], t^j), j=1..2*n+1) od; # yields sequence in triangular form

Crossrefs

Cf. A000108, A004148, A000984.

Sequence in context: A344937 A354061 A318921 * A218788 A328969 A027185

Adjacent sequences: A114513 A114514 A114515 * A114517 A114518 A114519

Keyword

nonn,tabf

Author

Emeric Deutsch, Dec 04 2005

Status

approved