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Number T(n,k) of modified skew Dyck paths of semilength n such that the area between the x-axis and the path is k; triangle T(n,k), n>=0, n<=k<=n^2, read by rows.
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%I #22 Jan 25 2017 03:05:55

%S 1,1,1,0,1,1,0,2,1,1,0,1,1,0,3,2,3,1,3,2,2,1,1,0,1,1,0,4,3,7,4,7,5,8,

%T 6,6,3,5,4,3,2,2,1,1,0,1,1,0,5,4,12,10,17,12,20,18,22,14,19,16,18,14,

%U 14,12,12,7,8,7,5,4,3,2,2,1,1,0,1

%N Number T(n,k) of modified skew Dyck paths of semilength n such that the area between the x-axis and the path is k; triangle T(n,k), n>=0, n<=k<=n^2, read by rows.

%C 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.

%H Alois P. Heinz, <a href="/A274372/b274372.txt">Rows n = 0..40, flattened</a>

%F Sum_{k=n..n^2} k * T(n,k) = A274373(n).

%F T(n,n) = T(n,n^2) = 1.

%F T(n,n+1) = T(n,n^2-1) = 0.

%F T(n,n*(n+1)/2) = T(n,A000217(n)) = A274054(n).

%e T(3,3) = 1: /\/\/\

%e .

%e /\ /\

%e T(3,5) = 2: /\/ \ , / \/\

%e .

%e /\

%e \ \

%e T(3,6) = 1: / \

%e .

%e /\/\

%e T(3,7) = 1: / \

%e .

%e /\

%e / \

%e T(3,9) = 1: / \

%e .

%e Triangle T(n,k) begins:

%e n\k: 0 1 2 3 4 5 6 7 8 9 . . . . . .16 . . . . . . . .25

%e ---+----------------------------------------------------

%e 00 : 1

%e 01 : 1

%e 02 : 1 0 1

%e 03 : 1 0 2 1 1 0 1

%e 04 : 1 0 3 2 3 1 3 2 2 1 1 0 1

%e 05 : 1 0 4 3 7 4 7 5 8 6 6 3 5 4 3 2 2 1 1 0 1

%p b:= proc(x, y, t, n) option remember; expand(`if`(y>n, 0,

%p `if`(n=y, `if`(t=2, 0, 1), b(x+1, y+1, 0, n-1)*z^

%p (2*y+1)+`if`(t<>1 and x>0, b(x-1, y+1, 2, n-1), 0)+

%p `if`(t<>2 and y>0, b(x+1, y-1, 1, n-1), 0))))

%p end:

%p T:= n-> (p-> seq(coeff(p, z, i), i=n..n^2))(b(0$3, 2*n)):

%p seq(T(n), n=0..8);

%t 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]*z^(2*y+1) + If[t != 1 && x>0, b[x-1, y+1, 2, n-1], 0] + If[t != 2 && y>0, b[x+1, y-1, 1, n-1], 0]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, n, n^2}]][b[0, 0, 0, 2*n]]; Table[T[n], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Jan 25 2017, translated from Maple *)

%Y Row sums give: A230823.

%Y Column sums give: A274376.

%Y Cf. A000217, A002061 (number of terms in row n), A129172, A274054, A274373.

%K nonn,tabf

%O 0,8

%A _Alois P. Heinz_, Jun 19 2016