A288972 -id:A288972 - cp's OEIS frontend

A176479 a(n) = (n+1)*A001003(n).

1, 2, 9, 44, 225, 1182, 6321, 34232, 187137, 1030490, 5707449, 31760676, 177435297, 994551222, 5590402785, 31500824304, 177880832001, 1006362234162, 5703029112297, 32367243171740, 183945502869345, 1046646207221582, 5961966567317649, 33995080211156904

Offset: 0

Paul Barry, Apr 18 2010

Central coefficients T(2n,n) of the Riordan array ((1-x)/(1-2x), x(1-x)/(1-2x)), A105306.
a(n) counts the bi-degree sequences of directed trees (i.e., digraphs whose underlying graph is a tree) with n edges. - Nikos Apostolakis, Dec 31 2016
a(n) is also the number of Dyck paths having exactly n peaks in level 1 and n peaks in level 2 and no other peaks. a(2) = 9: /\/\//\/\\, /\//\/\\/\, //\/\\/\/\, /\/\//\\//\\, /\//\\/\//\\, /\//\\//\\/\, //\\/\/\//\\, //\\/\//\\/\, //\\//\\/\/\. - Alois P. Heinz, Jun 20 2017
For n>0, a(n) is the number of ordered trees with n+1 leaves, no node of outdegree 1, and having one of its leaves marked. - Juan B. Gil, Jan 03 2024

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Sergi Elizalde, Rigoberto Flórez, and José Luis Ramírez, Enumerating symmetric peaks in non-decreasing Dyck paths, Ars Mathematica Contemporanea (2021).
Juan B. Gil, Emma G. Hoover, and Jessica A. Shearer, Bijections between colored compositions, Dyck paths, and polygon partitions, arXiv:2403.04575 [math.CO], 2024.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
V. V. Kruchinin and D. V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, arXiv preprint arXiv:1206.0877 [math.CO], 2012, and J. Int. Seq. 15 (2012) #12.9.3

Cf. A001003, A105306.

Row n=2 of A288972.

a:= proc(n) option remember; `if`(n<2, n+1,
     (6*n-3)/n*a(n-1) -(n-2)/(n-1)*a(n-2))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 22 2017

a[n_] := Sum[Binomial[n - 1, k - 1]*Binomial[n + k, n], {k, 0, n}]; Array[a, 25, 0] (* or *)
CoefficientList[ Series[1/4 - (x - 3)/(4 Sqrt[x^2 - 6x +1]), {x, 0, 25}], x] (* Robert G. Wilson v, Dec 31 2016 *)
Table[(n+1)Hypergeometric2F1[1-n, -n, 2, 2], {n,0,21}] (* Peter Luschny, Jan 02 2017 *)

E.g.f.: 1+exp(3*x)*Bessel_I(1,2*sqrt(2)*x)/sqrt(2) +int(exp(3*x) *Bessel_I(1,2*sqrt(2)*x) /(sqrt(2)*x),x).
G.f.: 1/4 - (x-3)/(4*sqrt(x^2-6*x+1)). - Dmitry Kruchinin, Aug 31 2012
Conjecture: n*(n-1)*a(n) -3*(2*n-1)*(n-1)*a(n-1) +n*(n-2)*a(n-2) = 0. - R. J. Mathar, Dec 03 2014
a(n) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(n+k,n). - Nikos Apostolakis, Dec 31 2016
a(n) = (n+1)*hypergeom([1-n, -n], [2], 2). - Peter Luschny, Jan 02 2017

A289020 Number of Dyck paths having exactly one peak in each of the levels 1,...,n and no other peaks.

Original entry on oeis.org

1, 1, 2, 10, 92, 1348, 28808, 845800, 32664944, 1605553552, 97868465696, 7245440815264, 640359291096512, 66598657958731840, 8051483595083729024, 1119653568781387712128, 177465810459239319017216, 31804047327185301634148608, 6398867435594240638421950976

Offset: 0

Alois P. Heinz, Jun 22 2017

nonn

The semilengths of Dyck paths counted by a(n) are elements of the integer interval [2*n-1, n*(n+1)/2] = [A060747(n), A000217(n)] for n>0.

			. a(2) = 2:      /\    /\
.             /\/  \  /  \/\  .

Alois P. Heinz, Table of n, a(n) for n = 0..100
Wikipedia, Counting lattice paths

Column k=1 of A288972.

Cf. A000217, A060747, A281874, A287846.

b:= proc(n, j, v) option remember; `if`(n=j,
      `if`(v=1, 1, 0), `if`(v<2, 0, add(b(n-j, i, v-1)*
       i*binomial(j-1, i-2), i=1..min(j+1, n-j))))
    end:
a:= n-> `if`(n=0, 1, add(b(w, 1, n), w=2*n-1..n*(n+1)/2)):
seq(a(n), n=0..18);

b[n_, j_, v_]:=b[n, j, v]=If[n==j, If[v==1, 1, 0], If[v<2, 0, Sum[b[n - j, i, v - 1]*i*Binomial[j - 1, i - 2], {i, Min[j + 1, n - j]}]]]; a[n_]:=If[n==0, 1, Sum[b[w, 1, n], {w, 2*n - 1, n*(n + 1)/2}]]; Table[a[n], {n, 0, 18}] (* Indranil Ghosh, Jul 06 2017, after Maple code *)

A288940 Number of Dyck paths having n (positive) levels and exactly n peaks per level.

Original entry on oeis.org

1, 1, 9, 27076, 147556480375, 4711342006036190504484, 2162932174406679548553402518043252929, 29605698225102450501737027784037791564430800582087459328, 22346336234943531646124131709622442581521043809236751640919325993842966011809319

Offset: 0

Alois P. Heinz, Jun 19 2017

nonn

The semilengths of Dyck paths counted by a(n) are elements of the integer interval [n^2+n-1, n^2*(n+1)/2] for n>0.

			. a(1) = 1:       /\  .
.
. a(2) = 9:           /\/\        /\/\        /\/\             /\  /\
.                /\/\/    \    /\/    \/\    /    \/\/\   /\/\/  \/  \
.
.    /\    /\      /\  /\      /\      /\    /\    /\      /\  /\
. /\/  \/\/  \  /\/  \/  \/\  /  \/\/\/  \  /  \/\/  \/\  /  \/  \/\/\ .

Alois P. Heinz, Table of n, a(n) for n = 0..15
Wikipedia, Counting lattice paths

Main diagonal of A288972.

Cf. A288318.

b:= proc(n, k, j, v) option remember; `if`(n=j, `if`(v=1, 1, 0),
      `if`(v<2, 0, add(b(n-j, k, i, v-1)*(binomial(i, k)
       *binomial(j-1, i-1-k)), i=1..min(j+k, n-j))))
    end:
a:= n-> `if`(n=0, 1, add(b(k, n$3), k=n^2+n-1..n^2*(n+1)/2)):
seq(a(n), n=0..7);

b[n_, k_, j_, v_]:=b[n, k, j, v]=If[n==j, If[v==1, 1, 0], If[v<2, 0, Sum[b[n - j, k, i, v - 1] Binomial[i, k] Binomial[j - 1, i - 1 - k], {i, Min[j + k, n - j]}]]]; a[n_]:=If[n==0, 1, Sum[b[k, n, n, n], {k, n^2 + n - 1, n^2*(n + 1)/2}]]; Table[a[n], {n, 0, 8}] (* Indranil Ghosh, Jul 06 2017, after Maple code *)

A289030 Number of Dyck paths having exactly n peaks in each of the levels 1,2,3 and no other peaks.

Original entry on oeis.org

1, 10, 471, 27076, 1713955, 114751470, 7969151855, 567878871304, 41247976697019, 3040572724077010, 226777538499783271, 17076122335343354700, 1296037531424347164115, 99025149551454886937590, 7609414766853344476768095, 587623058661705739915402256

Offset: 0

Alois P. Heinz, Jun 22 2017

nonn

The semilengths of Dyck paths counted by a(n) are elements of the integer interval [3*n+2, 6*n] for n>0.

			. a(1) = 10:
.
.        /\        /\          /\        /\
.     /\/  \      /  \/\    /\/  \      /  \/\
.  /\/      \  /\/      \  /      \/\  /      \/\
.
.                /\        /\                /\
.           /\  /  \      /  \  /\    /\    /  \
.        /\/  \/    \  /\/    \/  \  /  \/\/    \
.
.              /\        /\            /\
.         /\  /  \      /  \    /\    /  \  /\
.        /  \/    \/\  /    \/\/  \  /    \/  \/\  .

Alois P. Heinz, Table of n, a(n) for n = 0..523
Wikipedia, Counting lattice paths

Row n=3 of A288972.

b:= proc(n, k, j, v) option remember; `if`(n=j, `if`(v=1, 1, 0),
      `if`(v<2, 0, add(b(n-j, k, i, v-1)*(binomial(i, k)*
       binomial(j-1, i-1-k)), i=1..min(j+k, n-j))))
    end:
a:= n-> `if`(n=0, 1, add(b(w, n$2, 3), w=3*n+2..6*n)):
seq(a(n), n=0..15);

b[n_, k_, j_, v_]:=b[n, k, j, v]=If[n==j, If[v==1, 1, 0], If[v<2, 0, Sum[b[n - j, k, i, v - 1] Binomial[i, k] Binomial[j - 1, i - 1 - k], {i, Min[j + k, n - j]}]]]; a[n_]:=If[n==0, 1, Sum[b[w, n, n, 3], {w, 3n + 2, 6n}]]; Table[a[n], {n, 0, 15}] (* Indranil Ghosh, Jul 06 2017, after maple code *)

A289054 Number of Dyck paths having exactly two peaks in each of the levels 1,...,n and no other peaks.

Original entry on oeis.org

1, 1, 9, 471, 82899, 36913581, 34878248649, 62045165964951, 190543753640526939, 945931782247964900901, 7209377339218632463758129, 80920117567254715984058542191, 1292645840976784584918218615760819, 28557854803885245556927129118200208781

Offset: 0

Alois P. Heinz, Jun 23 2017

nonn

The semilengths of Dyck paths counted by a(n) are elements of the integer interval [3*n-1, n*(n+1)] for n>0.

			. a(2) = 9:           /\/\        /\/\        /\/\             /\  /\
.                /\/\/    \    /\/    \/\    /    \/\/\   /\/\/  \/  \
.
.    /\    /\      /\  /\      /\      /\    /\    /\      /\  /\
. /\/  \/\/  \  /\/  \/  \/\  /  \/\/\/  \  /  \/\/  \/\  /  \/  \/\/\ .

Alois P. Heinz, Table of n, a(n) for n = 0..60
Wikipedia, Counting lattice paths

Column k=2 of A288972.

b:= proc(n, j, v) option remember; `if`(n=j, `if`(v=1, 1, 0),
      `if`(v<2, 0, add(b(n-j, i, v-1)*(binomial(i, 2)*
       binomial(j-1, i-3)), i=1..min(j+2, n-j))))
    end:
a:= n-> `if`(n=0, 1, add(b(w, 2, n), w=3*n-1..n*(n+1))):
seq(a(n), n=0..15);

b[n_, j_, v_]:=b[n, j, v]=If[n==j, If[v==1, 1, 0], If[v<2, 0, Sum[b[n - j, i, v - 1] Binomial[i, 2] Binomial[j - 1, i - 3], {i, Min[j + 2, n - j]}]]]; a[n_]:=If[n==0, 1, Sum[b[w, 2, n], {w, 3*n - 1, n(n + 1)}]]; Table[a[n], {n, 0, 15}] (* Indranil Ghosh, Jul 06 2017, after Maple code *)

cp's OEIS Frontend

A176479 a(n) = (n+1)*A001003(n).

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A289020 Number of Dyck paths having exactly one peak in each of the levels 1,...,n and no other peaks.

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A288940 Number of Dyck paths having n (positive) levels and exactly n peaks per level.

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A289030 Number of Dyck paths having exactly n peaks in each of the levels 1,2,3 and no other peaks.

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A289054 Number of Dyck paths having exactly two peaks in each of the levels 1,...,n and no other peaks.

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