login
A274404
Number T(n,k) of modified skew Dyck paths of semilength n with exactly k anti-down steps; triangle T(n,k), n>=0, 0<=k<=n-floor((1+sqrt(max(0,8n-7)))/2), read by rows.
5
1, 1, 2, 5, 1, 14, 6, 42, 28, 3, 132, 120, 28, 1, 429, 495, 180, 20, 1430, 2002, 990, 195, 10, 4862, 8008, 5005, 1430, 165, 4, 16796, 31824, 24024, 9009, 1650, 117, 1, 58786, 125970, 111384, 51688, 13013, 1617, 70, 208012, 497420, 503880, 278460, 89180, 16016, 1386, 35
OFFSET
0,3
COMMENTS
A modified 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 A=(-1,1) (anti-down) so that A and D steps do not overlap.
LINKS
FORMULA
Sum_{k>0} k * T(n,k) = A274405(n).
EXAMPLE
/\
\ \
T(3,1) = 1: / \
.
Triangle T(n,k) begins:
: 1;
: 1;
: 2;
: 5, 1;
: 14, 6;
: 42, 28, 3;
: 132, 120, 28, 1;
: 429, 495, 180, 20;
: 1430, 2002, 990, 195, 10;
: 4862, 8008, 5005, 1430, 165, 4;
: 16796, 31824, 24024, 9009, 1650, 117, 1;
MAPLE
b:= proc(x, y, t, n) option remember; expand(`if`(y>n, 0,
`if`(n=y, `if`(t=2, 0, 1), b(x+1, y+1, 0, n-1)+
`if`(t<>1 and x>0, b(x-1, y+1, 2, n-1)*z, 0)+
`if`(t<>2 and y>0, b(x+1, y-1, 1, n-1), 0))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(0$3, 2*n)):
seq(T(n), n=0..14);
MATHEMATICA
b[x_, y_, t_, n_] := b[x, y, t, n] = Expand[If[y > n, 0,
If[n == y, If[t == 2, 0, 1], b[x + 1, y + 1, 0, n - 1] +
If[t != 1 && x > 0, b[x - 1, y + 1, 2, n - 1] z, 0] +
If[t != 2 && y > 0, b[x + 1, y - 1, 1, n - 1], 0]]]];
T[n_] := CoefficientList[b[0, 0, 0, 2n], z];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Mar 27 2021, after Alois P. Heinz *)
CROSSREFS
Columns k=0-3 give: A000108, A002694(n-1), A074922(n-2), A232224(n-3).
Row sums give A230823.
Last elements of rows give A092392(n-1) for n>0.
Sequence in context: A138159 A118919 A319120 * A101282 A263776 A145879
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Jun 20 2016
STATUS
approved