A129172 -id:A129172 - cp's OEIS frontend

A274372 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.

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

Offset: 0

Alois P. Heinz, Jun 19 2016

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.

			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

Alois P. Heinz, Rows n = 0..40, flattened

Row sums give: A230823.
Column sums give: A274376.
Cf. A000217, A002061 (number of terms in row n), A129172, A274054, A274373.

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);

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 *)

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).

A129173 Total area below all skew Dyck paths of semilength n.

Original entry on oeis.org

0, 1, 9, 58, 336, 1853, 9945, 52487, 273939, 1418567, 7303791, 37441560, 191287254, 974642943, 4955123955, 25146686730, 127424717400, 644873878895, 3260055588615, 16465301636090, 83092583965020, 419031686115875

Offset: 0

Emeric Deutsch, Apr 09 2007

nonn

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(2)=9 because the areas below the skew Dyck paths UDUD, UUDD and UUDL are 2, 4 and 3, respectively.

G. C. Greubel and Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (terms 0..300 from Vincenzo Librandi)
E. Deutsch, E. Munarini and S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
Emeric Deutsch, Emanuele Munarini and Simone Rinaldi, Skew Dyck paths, area, and superdiagonal bargraphs, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562, Table 1.

Cf. A002212, A129172.

a[0]:=1: a[1]:=1: a[2]:=9: for n from 3 to 25 do a[n]:=((11*n^2-20*n-6)*a[n-1]-5*(7*n^2-19*n+7)*a[n-2]+25*(n-1)*(n-3)*a[n-3])/(n+1)/(n-2) od: seq(a[n],n=0..25);

CoefficientList[Series[(1+x)*(1-3*x-Sqrt[1-6*x+5*x^2])/(2*x*(1-5*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

z='z +O('z^25); concat([0], Vec((1+z)*(1-3*z-sqrt(1-6*z+5*z^2))/(2*z*(1-5*z)))) \\ G. C. Greubel, Feb 10 2017

a(n) = Sum_{k=0,..,n^2} k*A129172(n,k).
a(n) - 5*a(n-1) = A002212(n) + A002212(n-1).
G.f.: (1+z)*(1-3*z-sqrt(1-6*z+5*z^2))/(2*z*(1-5*z)).
(n+1)(n-2)a(n)-(11n^2-20n-6)a(n-1)+5(7n^2-19n+7)a(n-2)-25(n-1)(n-3)a(n-3) = 0.
a(n) ~ 6*5^(n-1)*(1-sqrt(5)/sqrt(Pi*n)) . - Vaclav Kotesovec, Oct 20 2012

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A274372 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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