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A274372
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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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6
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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, 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, 14, 12, 12, 7, 8, 7, 5, 4, 3, 2, 2, 1, 1, 0, 1
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OFFSET
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0,8
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COMMENTS
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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.
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LINKS
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Alois P. Heinz, Rows n = 0..40, flattened
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FORMULA
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Sum_{k=n..n^2} k * T(n,k) = A274373(n).
T(n,n) = T(n,n^2) = 1.
T(n,n+1) = T(n,n^2-1) = 0.
T(n,n*(n+1)/2) = T(n,A000217(n)) = A274054(n).
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EXAMPLE
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T(3,3) = 1: /\/\/\
.
/\ /\
T(3,5) = 2: /\/ \ , / \/\
.
/\
\ \
T(3,6) = 1: / \
.
/\/\
T(3,7) = 1: / \
.
/\
/ \
T(3,9) = 1: / \
.
Triangle T(n,k) begins:
n\k: 0 1 2 3 4 5 6 7 8 9 . . . . . .16 . . . . . . . .25
---+----------------------------------------------------
00 : 1
01 : 1
02 : 1 0 1
03 : 1 0 2 1 1 0 1
04 : 1 0 3 2 3 1 3 2 2 1 1 0 1
05 : 1 0 4 3 7 4 7 5 8 6 6 3 5 4 3 2 2 1 1 0 1
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MAPLE
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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)*z^
(2*y+1)+`if`(t<>1 and x>0, b(x-1, y+1, 2, n-1), 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=n..n^2))(b(0$3, 2*n)):
seq(T(n), n=0..8);
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MATHEMATICA
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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 *)
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CROSSREFS
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Row sums give: A230823.
Column sums give: A274376.
Cf. A000217, A002061 (number of terms in row n), A129172, A274054, A274373.
Sequence in context: A261769 A005590 A142598 * A037800 A144411 A138253
Adjacent sequences: A274369 A274370 A274371 * A274373 A274374 A274375
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KEYWORD
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nonn,tabf
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AUTHOR
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Alois P. Heinz, Jun 19 2016
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STATUS
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approved
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