

A128741


Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k returns to the xaxis (1 <= k <= n).


0



1, 2, 1, 6, 3, 1, 20, 11, 4, 1, 72, 42, 17, 5, 1, 274, 166, 72, 24, 6, 1, 1086, 675, 307, 111, 32, 7, 1, 4438, 2809, 1322, 506, 160, 41, 8, 1, 18570, 11913, 5752, 2296, 775, 220, 51, 9, 1, 79174, 51319, 25274, 10418, 3692, 1127, 292, 62, 10, 1, 342738, 223977, 112054
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OFFSET

1,2


COMMENTS

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the xaxis, consists of steps U=(1,1)(up), D=(1,1)(down) and L=(1,1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps.
Row sums yield A002212.
T(n,1) = 2*A002212(n1) for n >= 2 (obvious: the path of semilength n with exactly one return are of the form UPD and UPL, where P is a path of semilength n1).
Sum_{k=1..n} k*T(n,k) = A128742(n).


LINKS

Table of n, a(n) for n=1..58.
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 21912203


FORMULA

G.f.: (1  tz + tzg)/(1  tzg)  1, where g = 1 + zg^2 + z(g1) = (1  z  sqrt(1  6z + 5z^2))/(2z).
Column k has g.f. z^k*g^(k1)*(2g1).


EXAMPLE

T(4,3)=4 because we have U(D)U(D)UUD(D), U(D)U(D)UUD(L), U(D)UUD(D)U(D) and UUD(D)U(D)U(D) (the return steps to the xaxis are shown between parentheses).
Triangle starts:
1;
2, 1;
6, 3, 1;
20, 11, 4, 1;
72, 42, 17, 5, 1;


MAPLE

g:=(1zsqrt(16*z+5*z^2))/2/z: G:=(1t*z+t*z*g)/(1t*z*g)1: Gser:=simplify(series(G, z=0, 15)): for n from 1 to 13 do P[n]:=sort(coeff(Gser, z, n), n=1..11) od: for n from 1 to 11 do seq(coeff(P[n], t, j), j=1..n) od;


CROSSREFS

Cf. A002212, A128742.
Sequence in context: A239103 A246971 A092392 * A175757 A060539 A163269
Adjacent sequences: A128738 A128739 A128740 * A128742 A128743 A128744


KEYWORD

tabl,nonn


AUTHOR

Emeric Deutsch, Mar 31 2007


STATUS

approved



