A274372 -id:A274372 - cp's OEIS frontend

A230823 Number of modified skew Dyck paths of semilength n.

1, 1, 2, 6, 20, 73, 281, 1124, 4627, 19474, 83421, 362528, 1594389, 7083078, 31738724, 143281473, 651048571, 2975243348, 13665866849, 63055369522, 292130900461, 1358415528683, 6337824891559, 29660089051015, 139193062791189, 654903798282528, 3088627236146085

Offset: 0

David Scambler and Alois P. Heinz, Oct 31 2013

nonn

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.

			a(0) = 1: the empty path.
a(1) = 1: UD.
a(2) = 2: UUDD, UDUD.
a(3) = 6: UUUDDD, UUDUDD, UUDDUD, UAUDDD, UDUUDD, UDUDUD.
a(4) = 20: UUUUDDDD, UUUDUDDD, UUUDDUDD, UUUDDDUD, UUAUDDDD, UUDUUDDD, UUDUDUDD, UUDUDDUD, UUDDUUDD, UUDDUDUD, UAUUDDDD, UAUDUDDD, UAUDDUDD, UAUDDDUD, UDUUUDDD, UDUUDUDD, UDUUDDUD, UDUAUDDD, UDUDUUDD, UDUDUDUD.
a(5) = 73: UUUUUDDDDD, UUUUDUDDDD, UUUUDDUDDD, ..., UDUDUAUDDD, UDUDUDUUDD, UDUDUDUDUD.

Alois P. Heinz, Table of n, a(n) for n = 0..600

Cf. A000108, A128714.

Row sums of A274372 and of A274404.

b:= proc(x, y, t, n) option remember; `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), 0)+
      `if`(t<>2 and y>0, b(x+1, y-1, 1, n-1), 0)))
    end:
a:= n-> b(0$3, 2*n):
seq(a(n), n=0..30);

b[x_, y_, t_, n_] := b[x, y, t, n] = 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], 0] + If[t != 2 && y > 0, b[x+1, y-1, 1, n-1], 0]] ]; a[n_] := b[0, 0, 0, 2*n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 16 2013, translated from Maple *)

a(n) ~ c * 5^n / n^(3/2), where c = 0.27726256768213709977373928535... . - Vaclav Kotesovec, Jul 16 2014

G.f.: 1/(1 - x/(1 - (x + x^2)/(1 - (x + x^2 + x^3)/(1 - (x + x^2 + x^3 + x^4)/(1 - ...))))), a continued fraction (conjecture). - Ilya Gutkovskiy, Jun 08 2017

A129172 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n such that the area between the x-axis and the path is k (n >= 0, 0 <= k <= n^2).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 3, 2, 1, 1, 1, 0, 0, 0, 0, 1, 1, 4, 5, 5, 3, 5, 4, 3, 2, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 5, 7, 12, 10, 11, 12, 14, 12, 10, 8, 10, 7, 5, 4, 3, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 6, 9, 19, 23, 28, 26, 36, 38, 38, 32, 36, 36, 34, 29, 27, 25, 21, 15, 16, 13, 10, 7, 5

Offset: 0

Emeric Deutsch, Apr 09 2007

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.
Row n has n^2 + 1 terms, the first n of which are equal to 0.
Row sums yield A002212.
Sum of terms in column k is the Fibonacci number F(k) (k >= 1; F(1)=1, F(2)=1; A000045).

			T(4,7)=5 because we have UDUUUDLD, UDUUDUDL, UUDDUUDL, UUUDLDUD and UUUUDLLL.
Triangle starts:
  1;
  0, 1;
  0, 0, 1, 1, 1;
  0, 0, 0, 1, 1, 3, 2, 1, 1, 1;
  0, 0, 0, 0, 1, 1, 4, 5, 5, 3, 5, 4, 3, 2, 1, 1, 1;

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

Cf. A000045, A002212, A129173, A274372.

G:=(1-z+z*g[1])/(1-t*x*z*g[1]): for i from 1 to 9 do g[i]:=(1-z+z*g[i+1])/(1-t^(2*i+1)*x*z*g[i+1]) od: g[10]:=0: x:=1: Gser:=simplify(series(G,z=0,9)): for n from 0 to 7 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 7 do seq(coeff(P[n],t,j),j=0..n^2) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y, t) option remember; expand(`if`(y>n, 0,
      `if`(n=0, 1, `if`(t<0, 0, b(n-1, y+1, 1)*z^(y+1/2))+
      `if`(y<1, 0, b(n-1, y-1, 0)*z^(y-1/2))+
      `if`(t>0 or y<1, 0, b(n-1, y-1, -1)*z^(1/2-y)))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..n^2))(b(2*n, 0$2)):
seq(T(n), n=0..8);  # Alois P. Heinz, Jun 19 2016

b[n_, y_, t_] := b[n, y, t] = Expand[If[y > n, 0, If[n == 0, 1, If[t < 0, 0, b[n - 1, y + 1, 1]*z^(y + 1/2)] + If[y < 1, 0, b[n - 1, y - 1, 0]*z^(y - 1/2)] + If[t > 0 || y < 1, 0, b[n - 1, y - 1, -1]*z^(1/2 - y)]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, n^2}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Dec 20 2016, after Alois P. Heinz *)

Sum_{k=0..n^2} k*T(n,k) = A129173(n).

G.f.: G(t,z) = H(t,1,z), where H(t,x,z) = 1+txzH(t,t^2*x,z)H(t,x,z) + z[H(t,t^2*x,z)-1] (H(t,x,z) is the trivariate g.f. for skew Dyck paths according to area, semiabscissa of the last point on the x-axis and semilength, marked by t,x and z, respectively).

A274373 Sum of the areas of all modified skew Dyck paths of semilength n.

Original entry on oeis.org

0, 1, 6, 35, 188, 989, 5131, 26411, 135229, 689814, 3509014, 17811637, 90256685, 456719880, 2308440442, 11656409995, 58809893357, 296500180806, 1493924791698, 7523064390774, 37866103978109, 190510720248534, 958122016323881, 4816944544836927, 24209532464417585

Offset: 0

Alois P. Heinz, Jun 19 2016

nonn

a(n)^(1/n) tends to 5. - Vaclav Kotesovec, Jun 26 2016

			a(3) = 35 = 9+7+5+6+5+3 = sum of the areas of UUUDDD, UUDUDD, UUDDUD, UAUDDD, UDUUDD, UDUDUD, respectively.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A230823, A274372.

b:= proc(x, y, t, n) option remember; `if`(y>n, 0, `if`(n=y,
     `if`(t=2, 0, [1, 0]), (p-> p+[0, p[1]*(2*y+1)])(b(x+1, y
      +1, 0, n-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:
a:= n-> b(0$3, 2*n)[2]:
seq(a(n), n=0..30);

b[x_, y_, t_, n_] := b[x, y, t, n] = If[y > n, 0, If[n == y, If[t == 2, {0, 0}, {1, 0}], Function[p, p + {0, p[[1]] (2y + 1)}][b[x + 1, y + 1, 0, n - 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]]];
a[n_] := b[0, 0, 0, 2 n][[2]];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)

a(n) = Sum_{k=n..n^2} k * A274372(n,k).

A274054 Number of modified skew Dyck paths of semilength n such that the area between the x-axis and the path is n*(n+1)/2.

Original entry on oeis.org

1, 1, 0, 1, 3, 6, 14, 40, 140, 422, 1346, 4487, 15234, 52632, 183913, 651948, 2336751, 8438406, 30712379, 112603500, 415459873, 1541646225, 5750112809, 21548036621, 81096740799, 306404247854, 1161863199131, 4420429256826, 16869986745367, 64567073382731

Offset: 0

Alois P. Heinz, Jun 19 2016

nonn

			a(1) = 1:   /\
.
            /\
            \ \
a(3) = 1:   /  \
.
                /\                          /\
               /  \         /\/\/\         /  \
a(4) = 3:   /\/    \   ,   /      \   ,   /    \/\   .

Cf. A000217, A230823, A274372.

a(n) = A274372(n,n*(n+1)/2) = A274372(n,A000217(n)).

A274376 Number of modified skew Dyck paths such that the area between the x-axis and the path is n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 8, 12, 20, 32, 52, 84, 135, 219, 353, 572, 924, 1495, 2419, 3912, 6331, 10240, 16570, 26807, 43374, 70178, 113546, 183721, 297258, 480974, 778220, 1259184, 2037389, 3296554, 5333923, 8630446, 13964340, 22594740, 36559034, 59153708, 95712668

Offset: 0

Alois P. Heinz, Jun 19 2016

nonn

			               /\    /\
a(5) = 3:   /\/  \  /  \/\  /\/\/\/\/\  .
.
            /\
            \ \    /\          /\          /\
a(6) = 5:   /  \  /  \/\/\  /\/  \/\  /\/\/  \  /\/\/\/\/\/\  .

Column sums of A274372.

Cf. A230823.

a(n) = Sum_{k=0..n} A274372(k,n).

cp's OEIS Frontend

A230823 Number of modified skew Dyck paths of semilength n.

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A129172 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n such that the area between the x-axis and the path is k (n >= 0, 0 <= k <= n^2).

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A274373 Sum of the areas of all modified skew Dyck paths of semilength n.

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Maple

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A274054 Number of modified skew Dyck paths of semilength n such that the area between the x-axis and the path is n*(n+1)/2.

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Comments

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A274376 Number of modified skew Dyck paths such that the area between the x-axis and the path is n.

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