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A129165
Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k base pyramids.
2
1, 0, 1, 1, 1, 1, 5, 2, 2, 1, 19, 9, 4, 3, 1, 73, 37, 15, 7, 4, 1, 292, 147, 63, 24, 11, 5, 1, 1203, 598, 258, 100, 37, 16, 6, 1, 5065, 2497, 1067, 419, 152, 55, 22, 7, 1, 21697, 10633, 4507, 1762, 647, 224, 79, 29, 8, 1, 94274, 45980, 19379, 7528, 2765, 964, 322
OFFSET
0,7
COMMENTS
A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, 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. A pyramid in a skew Dyck word (path) is a factor of the form u^h d^h, h being the height of the pyramid. A base pyramid is a pyramid starting on the x-axis.
Row sums yield A002212.
LINKS
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
FORMULA
T(n,0) = A129166(n).
Sum_{k=0..n} k*T(n,k) = A129167(n).
G.f.: G(t,z) = (1-z)(1 - z + zg)/(1 - z(1-z)g - tz), where g = 1 + zg^2 + z(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))(2z).
EXAMPLE
T(3,1)=2 because we have (UD)UUDL and (UUUDDD) (the base pyramids are shown between parentheses).
Triangle starts:
1;
0, 1;
1, 1, 1;
5, 2, 2, 1;
19, 9, 4, 3, 1;
MAPLE
g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=(1-z)*(1-z+z*g)/(1-z*(1-z)*g-t*z): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 11 do P[n]:=sort(expand(coeff(Gser, z, n))) od: for n from 0 to 11 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Apr 04 2007
STATUS
approved