A258172 -id:A258172 - cp's OEIS frontend

A258173 Sum over all Dyck paths of semilength n of products over all peaks p of y_p, where y_p is the y-coordinate of peak p.

1, 1, 3, 12, 58, 321, 1975, 13265, 96073, 743753, 6113769, 53086314, 484861924, 4641853003, 46441475253, 484327870652, 5252981412262, 59132909030463, 689642443691329, 8319172260103292, 103645882500123026, 1331832693574410475, 17629142345935969713

Offset: 0

Alois P. Heinz, May 22 2015

nonn

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Number of general rooted ordered trees with n edges and "back edges", which are additional edges connecting vertices to their ancestors. Every vertex specifies an ordering on the edges to its children and back edges to its ancestors altogether; it may be connected to the same ancestor by multiple back edges, distinguishable only by their relative ordering under that vertex. - Li-yao Xia, Mar 06 2017

Alois P. Heinz, Table of n, a(n) for n = 0..500
Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.
Antti Karttunen, Bijection between rooted trees with back edges and Dyck paths with multiplicity, SeqFans mailing list, Mar 2 2017.
Wikipedia, Lattice path
Li-yao Xia, Definition and enumeration of rooted trees with back edges in Haskell, blog post, Mar 1 2017.

Cf. A000108, A000698, A005411, A005412, A258172, A258174, A258175, A258176, A258177, A258178, A258179, A258180, A258181, A371567.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
     `if`(x=0, 1, b(x-1, y-1, 0)*y^t+b(x-1, y+1, 1)))
    end:
a:= n-> b(2*n, 0$2):
seq(a(n), n=0..25);

nmax = 25; Clear[g]; g[nmax+1] = 1; g[k_] := g[k] = 1 - x/(k*x + 2*x - 1/g[k+1]); CoefficientList[Series[g[0], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 20 2015, after Sergei N. Gladkovskii *)

G.f.: T(0), where T(k) = 1 - x/(k*x + 2*x - 1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 20 2015

Conjecture: a(n) = A371567(n-1,0) for n > 0 with a(0) = 1. - Mikhail Kurkov, Nov 07 2024

A258174 Sum over all Dyck paths of semilength n of products over all peaks p of x_p*y_p, where x_p and y_p are the coordinates of peak p.

Original entry on oeis.org

1, 1, 7, 84, 1486, 35753, 1111931, 43150593, 2035666985, 114412223081, 7538224510181, 574552299138202, 50096579094908148, 4949493445607316419, 549534510282406667069, 68071071679372210762156, 9347203754680124767253730, 1414740620049957735248175695

Offset: 0

Alois P. Heinz, May 22 2015

nonn

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Alois P. Heinz, Table of n, a(n) for n = 0..250
Wikipedia, Lattice path

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258175, A258176, A258177, A258178, A258179, A258180, A258181.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, x*y, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..20);

b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, x*y, 1] + b[x - 1, y + 1, True]]];
a[n_] := b[2*n, 0, False];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)

A258175 Sum over all Dyck paths of semilength n of products over all peaks p of x_p+y_p, where x_p and y_p are the coordinates of peak p.

Original entry on oeis.org

1, 2, 12, 114, 1448, 22770, 424164, 9095450, 220023184, 5914998594, 174682531260, 5614908340866, 194967208057272, 7267467723747218, 289270983756577620, 12239218862861690250, 548301077168477951520, 25918121712918957399426, 1288797080051656060595820

Offset: 0

Alois P. Heinz, May 22 2015

nonn

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Alois P. Heinz, Table of n, a(n) for n = 0..350
Wikipedia, Lattice path

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258176, A258177, A258178, A258179, A258180, A258181.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, x+y, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..20);

b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, x + y, 1] + b[x - 1, y + 1, True]]];
a[n_] := b[2*n, 0, False];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)

A258176 Sum over all Dyck paths of semilength n of products over all peaks p of x_p^y_p, where x_p and y_p are the coordinates of peak p.

Original entry on oeis.org

1, 1, 7, 142, 9354, 2503597, 3260627607, 24105227716863, 1028836978599566213, 290383808553140390346475, 511963364817949502725911280781, 6704846980724405836568589845161191576, 584709361918378923208855262622537662297053728

Offset: 0

Alois P. Heinz, May 22 2015

nonn

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Alois P. Heinz, Table of n, a(n) for n = 0..45
Wikipedia, Lattice path

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258177, A258178, A258179, A258180, A258181.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, x^y, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..15);

b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, x^y, 1] + b[x - 1, y + 1, True]]];
a[n_] :=  b[2*n, 0, False];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)

A258177 Sum over all Dyck paths of semilength n of products over all peaks p of y_p^x_p, where x_p and y_p are the coordinates of peak p.

Original entry on oeis.org

1, 1, 5, 112, 15312, 22928885, 475971133797, 164769697242392241, 1674694178196441599627207, 434453335415659344048321288040053, 2772047111897899211702422870954450438220795, 919691726760748842849028933552012720445531166591469510

Offset: 0

Alois P. Heinz, May 22 2015

nonn

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Alois P. Heinz, Table of n, a(n) for n = 0..35
Wikipedia, Lattice path

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258176, A258178, A258179, A258180, A258181.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, y^x, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..15);

b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, y^x, 1] + b[x - 1, y + 1, True]]];
a[n_] :=  b[2*n, 0, False];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)

A258178 Sum over all Dyck paths of semilength n of products over all peaks p of x_p^2, where x_p is the x-coordinate of peak p.

Original entry on oeis.org

1, 1, 13, 414, 24324, 2279209, 311524201, 58467947511, 14424374692879, 4525566110365523, 1759527523008436279, 830255082140922306224, 467382831980334193769718, 309419146352957449765072455, 237980526477430552734199922151, 210427994109788912088395561755374

Offset: 0

Alois P. Heinz, May 22 2015

nonn

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Alois P. Heinz, Table of n, a(n) for n = 0..200
Wikipedia, Lattice path

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258176, A258177, A258179, A258180, A258181.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, x^2, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..20);

b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, x^2, 1] + b[x - 1, y + 1, True] ]];
a[n_] :=  b[2*n, 0, False];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)

A258179 Sum over all Dyck paths of semilength n of products over all peaks p of y_p^2, where y_p is the y-coordinate of peak p.

Original entry on oeis.org

1, 1, 5, 34, 312, 3649, 52161, 889843, 17796555, 411120395, 10838039407, 322752018060, 10762432731362, 398802951148255, 16312276452291935, 732189190349581890, 35876807697443520000, 1910107567584518883891, 110035833179472385285367, 6832792252684597270659486

Offset: 0

Alois P. Heinz, May 22 2015

nonn

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Alois P. Heinz, Table of n, a(n) for n = 0..300
Wikipedia, Lattice path

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258176, A258177, A258178, A258180, A258181.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, y^2, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..20);

nmax = 20; Clear[g]; g[nmax+1] = 1; g[k_] := g[k] = 1 - x/( (k+2)^2*x - 1/g[k+1]); CoefficientList[Series[g[0], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 20 2015, after Sergei N. Gladkovskii *)

G.f.: T(0), where T(k) = 1 - x/( (k+2)^2*x - 1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 20 2015

A258180 Sum over all Dyck paths of semilength n of products over all peaks p of C(x_p,y_p), where x_p and y_p are the coordinates of peak p.

Original entry on oeis.org

1, 1, 4, 33, 517, 15326, 852912, 91023697, 19716262702, 8794395041567, 8016790849841585, 15556074485786226848, 64891787190080888991273, 561815453349204340865790817, 10402242033224422585780623039909, 423787530114579490372987256671625678

Offset: 0

Alois P. Heinz, May 22 2015

nonn

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Alois P. Heinz, Table of n, a(n) for n = 0..75
Wikipedia, Lattice path

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258176, A258177, A258178, A258179, A258181.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, binomial(x, y), 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..20);

b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, Binomial[x, y], 1] + b[x - 1, y + 1, True]]];
a[n_] := b[2*n, 0, False];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)

A258181 Sum over all Dyck paths of semilength n of products over all peaks p of 2^(x_p-y_p), where x_p and y_p are the coordinates of peak p.

Original entry on oeis.org

1, 1, 5, 89, 5933, 1540161, 1584150165, 6497470064169, 106497075348688637, 6980195689972655145233, 1829876050804408046228327525, 1918781572083632396857805205324025, 8047973452254281276702044410544321359565, 135022681866797995009325363468217320506328688097

Offset: 0

Alois P. Heinz, May 22 2015

nonn

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Alois P. Heinz, Table of n, a(n) for n = 0..55
Wikipedia, Lattice path

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258176, A258177, A258178, A258179, A258180.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, 2^(x-y), 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..15);

b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, 2^(x - y), 1] + b[x - 1, y + 1, True]]];
a[n_] := b[2*n, 0, False];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)

a(n) ~ c * 2^(n*(n-1)), where c = 1.47818066525747143617276638534... . - Vaclav Kotesovec, Jun 01 2015

cp's OEIS Frontend

A258173 Sum over all Dyck paths of semilength n of products over all peaks p of y_p, where y_p is the y-coordinate of peak p.

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A258174 Sum over all Dyck paths of semilength n of products over all peaks p of x_p*y_p, where x_p and y_p are the coordinates of peak p.

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A258175 Sum over all Dyck paths of semilength n of products over all peaks p of x_p+y_p, where x_p and y_p are the coordinates of peak p.

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A258176 Sum over all Dyck paths of semilength n of products over all peaks p of x_p^y_p, where x_p and y_p are the coordinates of peak p.

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Maple

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A258177 Sum over all Dyck paths of semilength n of products over all peaks p of y_p^x_p, where x_p and y_p are the coordinates of peak p.

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Maple

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A258178 Sum over all Dyck paths of semilength n of products over all peaks p of x_p^2, where x_p is the x-coordinate of peak p.

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Maple

Mathematica

A258179 Sum over all Dyck paths of semilength n of products over all peaks p of y_p^2, where y_p is the y-coordinate of peak p.

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Programs

Maple

Mathematica

Formula

A258180 Sum over all Dyck paths of semilength n of products over all peaks p of C(x_p,y_p), where x_p and y_p are the coordinates of peak p.

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