A026004 -id:A026004 - cp's OEIS frontend

A368567 Number of Young tableaux of shape [n, floor(n/2)].

1, 1, 2, 3, 9, 14, 48, 75, 275, 429, 1638, 2548, 9996, 15504, 62016, 95931, 389367, 600875, 2466750, 3798795, 15737865, 24192090, 100975680, 154969620, 650872404, 997490844, 4211628008, 6446369400, 27341497800, 41802112192, 177996090624, 271861216539, 1161588834303, 1772528290407, 7596549816030, 11582393855305

Offset: 0

Joerg Arndt, Dec 30 2023

nonn

Seemingly also the number of Catalan words of length n with at least ceiling(n/2) zeros. - Sela Fried, Jun 01 2025

Alois P. Heinz, Table of n, a(n) for n = 0..2417
Joerg Arndt, The a(6)=48 Young tableaux of shape [6,3] and a(7)=75 Young tableaux of shape [7,3].

Cf. A174687 (shape [2*n, n]), A026004 (shape [2*n+1, n]).

a:= proc(n) option remember; `if`(n<3, [1$2, 2][n+1],
      (4*n*(3027*n^2-10201*n+4134)*a(n-1)+6*(729*n^3-6201*n^2+9177*n-4921)*
      a(n-2)-3*(3*n-7)*(3027*n+1907)*(3*n-8)*a(n-3))/(8*(n+1)*n*(81*n-689)))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 01 2025

a(2*n) = A174687(n/2), a(2*n+1) = A026004(n).

A094021 Triangle read by rows: T(n,k) is the number of noncrossing forests with n vertices and k components (1<=k<=n).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 12, 14, 6, 1, 55, 75, 40, 10, 1, 273, 429, 275, 90, 15, 1, 1428, 2548, 1911, 770, 175, 21, 1, 7752, 15504, 13328, 6370, 1820, 308, 28, 1, 43263, 95931, 93024, 51408, 17640, 3822, 504, 36, 1, 246675, 600875, 648945, 406980, 162792, 42840

Offset: 1

Emeric Deutsch, May 31 2004

			From _Andrew Howroyd_, Nov 17 2017: (Start)
Triangle begins:
     1;
     1,     1;
     3,     3,     1;
    12,    14,     6,    1;
    55,    75,    40,   10,    1;
   273,   429,   275,   90,   15,   1;
  1428,  2548,  1911,  770,  175,  21,  1;
  7752, 15504, 13328, 6370, 1820, 308, 28, 1;
(End)
T(3,2)=3 because, with A,B,C denoting the vertices of a triangle, we have the 2-component forests (A,BC), (B,CA) and (C,AB).

Andrew Howroyd, Table of n, a(n) for n = 1..1275
P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math. 204 (1999), 203-229.

Columns k=1..2 are A001764, A026004.
Row sums are A054727.
Cf. A000108.

T:=proc(n,k) if k<=n then binomial(n,k-1)*binomial(3*n-2*k-1,n-k)/(2*n-k) else 0 fi end: seq(seq(T(n,k),k=1..n),n=1..11);

T[n_, k_] := If[k <= n, Binomial[n, k-1]*Binomial[3n-2k-1, n-k]/(2n-k), 0];
Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 06 2018 *)

T(n,k)=binomial(n, k-1)*binomial(3*n-2*k-1, n-k)/(2*n-k);
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 17 2017

T(n, k) = binomial(n, k-1)*binomial(3n-2k-1, n-k)/(2n-k).
G.f.: G=G(t, z) satisfies G^3+(t^3*z^2-t^2*z-3)G^2+(t^2*z+3)G-1=0.
From Peter Bala, Nov 07 2015: (Start)
O.g.f. A(x,t) = revert( x/((1 + x*t)*C(x)) ) with respect to x, where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f for the Catalan numbers A000108.
Row sums are A054727. (End)

A045722 Number of border edges in all noncrossing rooted trees on n nodes.

Original entry on oeis.org

1, 6, 28, 150, 858, 5096, 31008, 191862, 1201750, 7597590, 48384180, 309939240, 1994981688, 12892738800, 83604224384, 543722433078, 3545056580814, 23164787710610, 151662849838500, 994674967479270, 6533629880128890

Offset: 1

Emeric Deutsch

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..1208
Index entries for sequences related to rooted trees

Cf. A026004, A006013.

MapAt[# - 1 &, Array[(# + 1) Binomial[3 # - 2, # - 1]/# &, 21], 1] (* Michael De Vlieger, Mar 19 2021 *)

a(n) = (n+1)*binomial(3n-2, n-1)/n for n >= 2. [Corrected by Sean A. Irvine, Mar 19 2021]
G.f.: (1+g-7*g^2+3*g^3)/((1-3*g)*(g-1)^2) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011
D-finite with recurrence 2*n*(2*n-1)*a(n) + (-43*n^2+83*n-34)*a(n-1) + 12*(3*n-5)*(3*n-7)*a(n-2) = 0. - R. J. Mathar, Jul 26 2022
a(n) = (n+1) * A006013(n-1) for n >= 2. - F. Chapoton, Feb 27 2024

A094040 Triangle read by rows: T(n,k) is the number of noncrossing forests with n vertices and k edges.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 6, 14, 12, 1, 10, 40, 75, 55, 1, 15, 90, 275, 429, 273, 1, 21, 175, 770, 1911, 2548, 1428, 1, 28, 308, 1820, 6370, 13328, 15504, 7752, 1, 36, 504, 3822, 17640, 51408, 93024, 95931, 43263, 1, 45, 780, 7350, 42840, 162792, 406980, 648945, 600875, 246675

Offset: 1

Emeric Deutsch, May 31 2004

T(n,n-1) yields A001764; T(n,n-2) yields A026004.

			Triangle begins:
  1;
  1,  1;
  1,  3,   3;
  1,  6,  14,  12;
  1, 10,  40,  75,   55;
  1, 15,  90, 275,  429,  273;
  1, 21, 175, 770, 1911, 2548, 1428;
  ...
T(3,1)=3 because the noncrossing forests on 3 vertices A,B,C and having one edge are (A, BC), (B, CA) and (C, AB).

Andrew Howroyd, Table of n, a(n) for n = 1..1275
P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math. 204 (1999), 203-229.

Cf. A001764, A026004, A045739, A094021.

T:=proc(n,k) if k<=n-1 then binomial(n,k+1)*binomial(n+2*k-1,k)/(n+k) else 0 fi end: seq(seq(T(n,k),k=0..n-1),n=1..11);

T[n_, k_] := Binomial[n, k+1] Binomial[n+2k-1, k]/(n+k);
Table[T[n, k], {n, 1, 11}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jul 29 2018 *)

T(n,k)=binomial(n, k+1)*binomial(n+2*k-1, k)/(n+k);
for(n=1, 10, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 17 2017

T(n, k)=binomial(n, k+1)*binomial(n+2k-1, k)/(n+k) (0<=k<=n-1).

A277956 a(n) = (n+2)Sum_{i=0..n}(binomial(3n-2i+1, n-i)/(2n-i+2)).

Original entry on oeis.org

1, 4, 19, 101, 573, 3382, 20483, 126292, 788878, 4976489, 31635811, 202354517, 1300880374, 8398175713, 54409200963, 353571026085, 2303666554659, 15043760670031, 98439176169692, 645290365460761, 4236768489465944, 27857102370774193

Offset: 0

Vladimir Kruchinin, Nov 05 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A001764, A026004, A242798.

h := n -> hypergeom([1,-2*n-2,-n],[-3*n/2-1/2,-3*n/2],1/4):
b := n -> binomial(3*n+1,n)*(n+2)/(2*n+2): # A026004
a := n -> `if`(n=0,1,b(n)*simplify(h(n))):
seq(a(n), n=0..21); # Peter Luschny, Nov 06 2016

f[n_] := (n + 2)Sum[ Binomial[3n - 2i + 1, n - i]/(2n - i + 2), {i, 0, n}]; Array[f, 22, 0] (* Robert G. Wilson v, Nov 06 2016 *)

F(x):=x*(2/sqrt(3*x))*sin((1/3)*asin(sqrt(27*x/4)));
taylor(diff(F(x),x)*F(x)/(1-F(x))/x,x,0,10);

for(n=0,25, print1((n+2)*sum(i=0,n,(binomial(3*n-2*i+1, n-i)/(2*n-i+2))), ", ")) \\ G. C. Greubel, Apr 09 2017

G.f.: F'(x)*F(x)/(1-F(x))/x, where F(x)/x is g.f. of A001764.
From Vaclav Kotesovec, Nov 06 2016: (Start)
Recurrence: 2*(n+1)*(2*n + 1)*(91*n^4 - 232*n^3 + 15*n^2 + 266*n - 120)*a(n) = (2821*n^6 - 4189*n^5 - 10027*n^4 + 18573*n^3 - 3498*n^2 - 3968*n + 960)*a(n-1) - (2821*n^6 - 4189*n^5 - 10027*n^4 + 18573*n^3 - 3498*n^2 - 3968*n + 960)*a(n-2) + 3*(3*n - 5)*(3*n - 4)*(91*n^4 + 132*n^3 - 135*n^2 - 36*n + 20)*a(n-3).
a(n) ~ 3^(3*n+7/2) / (7 * sqrt(Pi*n) * 2^(2*n+3)). (End)
a(n) = A026004(n)*hypergeom([1,-2*n-2,-n],[-3*n/2-1/2,-3*n/2],1/4). - Peter Luschny, Nov 06 2016

A139816 Final nonzero terms in rows of A139801.

Original entry on oeis.org

1, 2, 3, 9, 14, 48, 75, 275, 429, 1638

Offset: 0

Paul Curtz, Jun 14 2008

nonn

First bisection a(2n) = 1, 3, 14, 75, 429, ... =A026004.

cp's OEIS Frontend

A368567 Number of Young tableaux of shape [n, floor(n/2)].

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A094021 Triangle read by rows: T(n,k) is the number of noncrossing forests with n vertices and k components (1<=k<=n).

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A045722 Number of border edges in all noncrossing rooted trees on n nodes.

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A094040 Triangle read by rows: T(n,k) is the number of noncrossing forests with n vertices and k edges.

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A277956 a(n) = (n+2)*Sum_{i=0..n}(binomial(3*n-2*i+1, n-i)/(2*n-i+2)).

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A139816 Final nonzero terms in rows of A139801.

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A277956 a(n) = (n+2)Sum_{i=0..n}(binomial(3n-2i+1, n-i)/(2n-i+2)).