A191793 Triangle read by rows: T(n,k) is the number of length n left factors of Dyck paths having k UUDD's, where U=(1,1) and D=(1,-1).

1, 1, 2, 3, 5, 1, 8, 2, 14, 6, 23, 12, 41, 28, 1, 69, 54, 3, 124, 116, 12, 212, 220, 30, 383, 453, 87, 1, 662, 852, 198, 4, 1200, 1712, 500, 20, 2091, 3204, 1080, 60, 3799, 6338, 2526, 206, 1, 6661, 11824, 5280, 540, 5, 12122, 23136, 11772, 1560, 30, 21359, 43068, 24066, 3780, 105

Offset

0,3

Comments

Row n contains 1+floor(n/4) entries.

Sum of entries in row n is binomial(n, floor(n/2)) =A001405(n).

T(n,0)=A191794(n).

Sum(k*T(n,k), k>=0)=A100071(n-3).

Links

Table of n, a(n) for n=0..59.

Formula

G.f.: G(t,z)= C/(1-z*C), where C=C(t,z) is given by z^2*C^2-(1+z^4-t*z^4)*C +1=0.

Example

T(9,2)=3 because we have (UUDD)(UUDD)U, (UUDD)U(UUDD), and U(UUDD)(UUDD), where U=(1,1) and D=(1,-1) (the UUDD's are shown between parentheses).
Triangle starts:
1;
1;
2;
3;
5,1;
8,2;
14,6;
23,12;
41,28,1;

Maple

eq := z^2*C^2-(1+z^4-t*z^4)*C+1 = 0: C := RootOf(eq, C): G := C/(1-z*C): Gser := simplify(series(G, z = 0, 23)): for n from 0 to 19 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 1; for n from 2 to 19 do seq(coeff(P[n], t, k), k = 0 .. floor((1/4)*n)) end do; # yields sequence in triangular form

Crossrefs

Cf. A001405, A191794

Sequence in context: A191399 A191316 A191387 * A191791 A132597 A030335

Adjacent sequences: A191790 A191791 A191792 * A191794 A191795 A191796

Keyword

nonn,tabf

Author

Emeric Deutsch, Jun 18 2011

Status

approved