A143950 Triangle read by rows: T(n,k) is the number of Dyck n-paths containing k even-length ascents (0 <= k <= floor(n/2)).

1, 1, 1, 1, 2, 3, 5, 7, 2, 12, 20, 10, 30, 61, 36, 5, 79, 182, 133, 35, 213, 547, 488, 168, 14, 584, 1668, 1728, 756, 126, 1628, 5116, 6020, 3240, 750, 42, 4600, 15752, 20812, 13200, 3960, 462, 13138, 48709, 71376, 52030, 19360, 3267, 132, 37871, 151164

Offset

0,5

Comments

Row n contains 1 + floor(n/2) entries.

Row sums are the Catalan numbers (A000108).

T(n,0) = A101785(n).

Sum_{k=0..floor(n/2)} k*T(n,k) = A014301(n).

For the Dyck path statistic "number of odd-length ascents" see A096793.

Links

Table of n, a(n) for n=0..50.

Formula

G.f. G=G(s,z) satisfies G = 1 + zG(1 + szG)/(1 - z^2*G^2).

The trivariate g.f. H=H(t,s,z), where t (s) marks odd-length (even-length) ascents satisfies H = 1 + zH(t+szH)/(1-z^2*H^2).

Example

T(4,1)=7 because we have UDUD(UU)DD, UD(UU)DDUD, UD(UU)DUDD, (UU)DDUDUD, (UU)DUDDUD, (UU)DUDUDD and (UUUU)DDDD (the even-length ascents are shown between parentheses).
Triangle starts:
   1;
   1;
   1,  1;
   2,  3;
   5,  7,  2;
  12, 20, 10;
  30, 61, 36,  5;

Maple

eq:=G=1+(1+s*z*G)*z*G/(1-z^2*G^2): G:=RootOf(eq, G): Gser:=simplify(series(G, z =0, 16)): for n from 0 to 13 do P[n]:=sort(expand(coeff(Gser, z, n))) end do: for n from 0 to 13 do seq(coeff(P[n], s, j), j=0..floor((1/2)*n)) end do; # yields sequence in triangular form

Crossrefs

Cf. A000108, A014301, A096793, A101785.

Sequence in context: A004088 A126051 A115260 * A108534 A039706 A156208

Adjacent sequences: A143947 A143948 A143949 * A143951 A143952 A143953

Keyword

nonn,tabf

Author

Emeric Deutsch, Oct 05 2008

Status

approved