A005789 -id:A005789 - cp's OEIS frontend

A268538 a(n) is the n-th prime 3-dimensional Catalan number.

1, 1, 2, 12, 107, 1178, 14805, 203885, 3002973, 46573347, 752521980, 12571607865, 215925120675, 3796546970232, 68106673339365, 1243210765414512, 23041656826384341

Offset: 0

N. J. A. Sloane, Feb 24 2016

nonn

"Prime" here is used in the sense of "primitive" or "irreducible".

Manuel Wettstein, Trapezoidal Diagrams, Upward Triangulations, and Prime Catalan Numbers, arXiv:1602.07235 [cs.CG], 2016.

Primitive terms from A000108, A005789.

A005789 := proc(n)
    2*(3*n)!/(n+2)!/(n+1)!/n! ;
end proc:
maxn := 30 :
Cx := add(A005789(i)*x^i,i=0..maxn) ;
d := 3:
for i from 0 to maxn do
    coeftayl(1/Cx^(d*i-1),x=0,i) ;
    %/(1-d*i) ;
    printf("%d,",%) ;
end do: # R. J. Mathar, Feb 27 2018

A005789[n_] := 2*(3*n)!/(n+2)!/(n+1)!/n!; Maxn = 30; Cx = Sum[A005789[i]* x^i, {i, 0, Maxn}]; d = 3; Reap[For[i = 0, i <= Maxn, i++, sc = SeriesCoefficient[1/Cx^(d*i-1), {x, 0, i}]; Sow[sc/(1-d*i)]]][[2, 1]] (* Jean-François Alcover, Mar 24 2018, after R. J. Mathar *)

Lemma 15 of Wettstein (2016) gives a formula in terms of the 3-dimensional Catalan numbers (A005789).

7 more terms from R. J. Mathar, Feb 27 2018

A321716 Triangle read by rows: T(n,k) is the number of n X k Young tableaux, where 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 5, 42, 1, 1, 14, 462, 24024, 1, 1, 42, 6006, 1662804, 701149020, 1, 1, 132, 87516, 140229804, 396499770810, 1671643033734960, 1, 1, 429, 1385670, 13672405890, 278607172289160, 9490348077234178440, 475073684264389879228560

Offset: 0

Seiichi Manyama, Nov 17 2018

			T(4,3) = 12! / ((6*5*4)*(5*4*3)*(4*3*2)*(3*2*1)) = 462.
Triangle begins:
  1;
  1, 1;
  1, 1,   2;
  1, 1,   5,    42;
  1, 1,  14,   462,     24024;
  1, 1,  42,  6006,   1662804,    701149020;
  1, 1, 132, 87516, 140229804, 396499770810, 1671643033734960;

Seiichi Manyama, Rows n = 0..30, flattened
Wikipedia, Hook length formula
Index entries for sequences related to Young tableaux.

Cf. A000178, A005789, A005790, A005791, A039622, A060854

A321716:= func< n,k | n eq 0 select 1 else Factorial(n*k)/(&*[ Round(Gamma(j+k)/Gamma(j)): j in [1..n]]) >;
[A321716(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 04 2021

T[n_, k_]:= (n*k)!/Product[Product[i+j-1, {j,1,k}], {i,1,n}]; Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Amiram Eldar, Nov 17 2018 *)
T[n_, k_]:= (n*k)!*BarnesG[n+1]*BarnesG[k+1]/BarnesG[n+k+1];
Table[T[n, k], {n, 0, 5}, {k, 0, n}] //Flatten (* G. C. Greubel, May 04 2021 *)

def A321716(n,k): return factorial(n*k)/product( gamma(j+k)/gamma(j) for j in (1..n) )
flatten([[A321716(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 04 2021

T(n, k) = (n*k)! / (Product_{i=1..n} Product_{j=1..k} (i+j-1)).
T(n, k) = A060854(n,k) for n,k > 0.
T(n, n) = A039622(n).
T(n, k) = (n*k)!*BarnesG(n+1)*BarnesG(k+1)/BarnesG(n+k+1), where BarnesG(n) = A000178. - G. C. Greubel, May 04 2021

A321978 9-dimensional Catalan numbers.

Original entry on oeis.org

1, 1, 4862, 414315330, 177295473274920, 219738059326729823880, 583692803893929928888544400, 2760171874087743799855959353857200, 20535535214275361308250745082811167425600, 220381378415074546123953914908618547085974856000

Offset: 0

Seiichi Manyama, Nov 23 2018

nonn

Number of n X 9 Young tableaux.

Seiichi Manyama, Table of n, a(n) for n = 0..124
Wikipedia, Hook length formula

Row 9 of A060854.

Cf. A000108, A005789, A005790, A005791, A321975, A321976, A321977.

List([0..10],n->5056584744960000*Factorial(9*n)/Product([0..8],k->Factorial(n+k))); # Muniru A Asiru, Nov 25 2018

[5056584744960000*Factorial(9*n)/(Factorial(n)*Factorial(n + 1)*Factorial(n + 2)*Factorial(n + 3)*Factorial(n + 4)*Factorial(n + 5)*Factorial(n + 6)*Factorial(n + 7)*Factorial(n + 8)): n in [0..15]]; // Vincenzo Librandi, Nov 24 2018

Table[5056584744960000 (9 n)! / (n! (n + 1)! (n + 2)! (n + 3)! (n + 4)! (n + 5)! (n + 6)! (n + 7)! (n + 8)!), {n, 0, 15}] (* Vincenzo Librandi, Nov 24 2018 *)

a(n) = 0!*1!*...*8! * (9*n)! / ( n!*(n+1)!*...*(n+8)! ).

a(n) ~ 16056320000 * 3^(18*n + 10) / (Pi^4 * n^40). - Vaclav Kotesovec, Nov 23 2018

A361190 Number of 4n-step lattice paths starting and ending at (0,0) that do not go above the diagonal x=y or below the x-axis using steps in {(1,1), (1,-1), (-1,0)}.

Original entry on oeis.org

1, 1, 9, 153, 3579, 101630, 3288871, 116951012, 4465824585, 180310624841, 7614208325878, 333613510494834, 15075162152856423, 699290488810583617, 33176816563410874752, 1605135467691243954419, 79003021319962788395355, 3947913343912428255683930

Offset: 0

Alois P. Heinz, Jul 31 2023

Is this the same sequence as A217823?

			a(0) = 1: (00), 0 steps are made.
a(1) = 1: (00)(11)(20)(10)(00).
a(2) = 9:
  (00)(11)(20)(10)(00)(11)(20)(10)(00),
  (00)(11)(20)(10)(21)(30)(20)(10)(00),
  (00)(11)(20)(10)(21)(11)(20)(10)(00),
  (00)(11)(20)(31)(40)(30)(20)(10)(00),
  (00)(11)(20)(31)(21)(30)(20)(10)(00),
  (00)(11)(20)(31)(21)(11)(20)(10)(00),
  (00)(11)(22)(31)(40)(30)(20)(10)(00),
  (00)(11)(22)(31)(21)(30)(20)(10)(00),
  (00)(11)(22)(31)(21)(11)(20)(10)(00).

Alois P. Heinz, Table of n, a(n) for n = 0..400
Wikipedia, Lattice path

Cf. A001006, A005789, A026945, A151332 (the same without condition on the diagonal), A217823, A359647.

b:= proc(n, x, y) option remember; `if`(x+2*y>n, 0,
     `if`(n=0, 1, `if`(y>0, b(n-1, x+1, y-1), 0)+
     `if`(y b(4*n, 0$2):
seq(a(n), n=0..17);

a(n) = A217823(n) for n<=6.

A065077 Triangle read by rows: T(n,m) = C[n,m,m] where C[i,j,k] is the 3-dimensional Catalan pyramid defined by C[0,0,0]=1 and C[i,j,k]=0 if j>i or k>j and C[i,j,k]=C[i-1,j,k]+C[i,j-1,k]+C[i,j,k-1].

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 1, 6, 21, 42, 1, 10, 56, 210, 462, 1, 15, 120, 660, 2574, 6006, 1, 21, 225, 1650, 9009, 36036, 87516, 1, 28, 385, 3575, 25025, 136136, 554268, 1385670, 1, 36, 616, 7007, 60060, 408408, 2217072, 9145422, 23371634, 1, 45, 936, 12740, 129948

Offset: 0

Wouter Meeussen, Nov 09 2001

T(n,m)= number of standard tableaux of shape (n,m,m) (0Emeric Deutsch, May 14 2004

			1;
1,1;
1,3,5;
1,6,21,42;
1,10,56,210,462;
1,15,120,660,2574,6006;
...
T(2,1)=3 because in the first row of the diagram (2,1,1) we can have 12 or 13 or 14.

Last number in each row is A005789

Cf. A005789, A065058, A065078.

a:=proc(n,m) if m<=n then (n+2*m)!*(n-m+1)*(n-m+2)/m!/(m+1)!/(n+2)! else 0 fi end: seq(seq(a(n,m),m=0..n),n=0..9);

C[0, 0, 0] := 1; C[x_, y_, z_] := 0 /; (x< y || y< z); C[u_, v_, 0] := (u+v)!/(u+1)!/(v)!(u-v+1); C[, 0, 0] := 1; C[x, y_, z_] := (C[x, y, z]= C[x-1, y, z]+C[x, y-1, z] +C[x, y, z-1]) /; (y<=x ||z<=y); Table[C[x, y, y], {x, 0, 10}, {y, 0, x}]

T(n, m)=(n+2m)!(n-m+1)(n-m+2)/[m!(m+1)!(n+2)! ] (0<=m<=n). - Emeric Deutsch, May 14 2004

A087727 Triangle read by rows: T(n,k) = (2 * (binomial(n,k)) * (n + 2 * k + 3)!)/((k + 1)! * (k + 2)! * (n + 3)!).

Original entry on oeis.org

1, 1, 5, 1, 14, 42, 1, 28, 210, 462, 1, 48, 660, 3432, 6006, 1, 75, 1650, 15015, 60060, 87516, 1, 110, 3575, 50050, 340340, 1108536, 1385670, 1, 154, 7007, 140140, 1429428, 7759752, 21339318, 23371634, 1, 208, 12740, 346528, 4938024, 39504192, 178474296, 424938800, 414315330

Offset: 0

Bill Gosper, Sep 30 2003

			Triangle begins:
  1;
  1,  5;
  1, 14,  42;
  1, 28, 210, 462;
  ...

Row sums give A087659. Right diagonal gives A005789.

t(n, k) = (2 * (binomial(n,k)) * (n + 2*k + 3)!)/((k + 1)! * (k + 2)! * (n + 3)!) \\ Michel Marcus, Jun 26 2013

More terms from Ray Chandler, Sep 30 2003

A123691 a(n) = number of standard Young tableaux of type (n,n-1,n-1).

Original entry on oeis.org

1, 3, 21, 210, 2574, 36036, 554268, 9145422, 159352050, 2900207310, 54698315490, 1062710129520, 21172455657360, 431010704453400, 8939669081780520, 188478023140872630, 4031562420682009290, 87350519114776867950, 1914486941500560677250, 42397183540866961907100

Offset: 1

Robert G. Wilson v, Nov 15 2006

Alois P. Heinz, Table of n, a(n) for n = 1..700
Wolfgang Unger, Combinatorics of Lattice QCD at Strong Coupling, arXiv:1411.4493 [hep-lat], 2014.

Cf. A005789, A123555, subdiagonal of A065077.

a:= n-> 6 *(3*n-2)! / (n! *(n-1)! *(n+2)!):
seq(a(n), n=1..25); # Alois P. Heinz, Apr 11 2012

(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) Table[ NumberOfTableaux@{n, n - 1, n - 1}, {n, 18}]

a(n) = 6*(3*n-2)! / (n!*(n-1)!*(n+2)!). - Alois P. Heinz, Apr 11 2012
n*(n+2)*a(n) - 3*(3*n-2)*(3*n-4)*a(n-1) = 0. - R. J. Mathar, Aug 10 2015
G.f.: x*3F2(1,2/3,4/3;2,4;27x). - R. J. Mathar, Aug 10 2015

A364439 a(n) is the number of paths with length 3*n that begin at (0,0), end at (0,0), and do not reach (0,0) at any point in between while 0 <= y <= x at every step, where a path is a sequence of steps in the form (1,1), (1,-1), and (-2,0).

Original entry on oeis.org

1, 1, 4, 33, 367, 4844, 71597, 1147653, 19559062, 349766457, 6502419671, 124822220086, 2461515013103, 49668479230825, 1022258042480874, 21406231023989503, 455112508356168561, 9807294681518154334, 213897254891041613995, 4715809234441541498539

Offset: 0

Marshall Hamon, Jul 24 2023

If the constraint is removed that the sequence does not reach (0,0) at any point other than the beginning and end of the sequence, this sequence becomes A005789.

			Let A represent the (1,1) step, B represent the (1,-1) step, and C represent the (-2,0) step.
For n = 1, the only valid path is ABC.
For n = 2, the 4 valid paths are AABBCC, AABCBC, ABABCC, ABACBC.

Alois P. Heinz, Table of n, a(n) for n = 0..707
Marshall Hamon, A364439.c

Cf. A005789, A011655.

C
```
/* See Hamon Link */
```

b:= proc(n, l) option remember; `if`(n<1, 1, add((h->
      `if`(h[2]>h[1] or h[1]>=n or min(h)<0 or n>1 and h=[0$2],
           0, b(n-1, h)))(l-w), w=[[1, 1], [1, -1], [-2, 0]]))
    end:
a:= n-> b(3*n-1, [2, 0]):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 28 2023
# second Maple program:
f:= proc(n) option remember; (3*n)!*mul(i!/(n+i)!, i=0..2) end:	
a:= proc(n) option remember; `if`(n=0, 1,
      f(n)-add(f(n-i)*a(i), i=1..n-1))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 29 2023

f[n_] := f[n] = (3n)!*Product[i!/(n+i)!, {i, 0, 2}];
a[n_] := a[n] = If[n == 0, 1, f[n] - Sum[f[n-i]*a[i], {i, 1, n-1}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 27 2023, after Alois P. Heinz *)

From Alois P. Heinz, Jul 29 2023: (Start)
INVERTi transform of A005789.
a(n) mod 2 = A011655(n+1). (End)

More terms from Alois P. Heinz, Jul 27 2023

A134236 Ternary numbers with equal numbers of 0's, 1's and 2's and such that read left-to-right at all times the number of 2's >= number of 1's >= number of 0's.

Original entry on oeis.org

210, 210210, 212010, 212100, 221010, 221100, 210210210, 210212010, 210212100, 210221010, 210221100, 212010210, 212012010, 212012100, 212021010, 212021100, 212100210, 212102010, 212102100, 212120010, 212120100, 212121000, 212201010

Offset: 1

Rick L. Shepherd, Oct 14 2007

A005789(n) counts the terms here with 3n ternary digits.

Cf. A005789, A063171, A000108.

A338368 Triangle of number of 312-avoiding reduced valid hook configurations with n hooks that are on permutations with 2n + k points, for n=1 and 1<=k<=n.

Original entry on oeis.org

1, 3, 5, 14, 51, 42, 84, 485, 849, 462, 594, 4743, 13004, 14819, 6006, 4719, 48309, 182311, 322789, 271452, 87516, 40898, 511607, 2472322, 5999489, 7794646, 5182011, 13856

Offset: 1

Michel Marcus, Oct 23 2020

			Triangle begins
  1
  3 5
  14 51 42
  84 485 849 462
  594 4743 13004 14819 6006
  4719 48309 182311 322789 271452 87516
  40898 511607 2472322 5999489 7794646 5182011 13856

Ilani Axelrod-Freed, 312-Avoiding Reduced Valid Hook Configurations and Duck Words, arXiv:2010.11834 [math.CO], 2020.
Maya Sankar, Further Bijections to Pattern-Avoiding Valid Hook Configurations, arXiv:1910.08895 [math.CO], 2019.

Cf. A005700 (1st column), A005789 (right diagonal).

cp's OEIS Frontend

A268538 a(n) is the n-th prime 3-dimensional Catalan number.

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A321716 Triangle read by rows: T(n,k) is the number of n X k Young tableaux, where 0 <= k <= n.

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A321978 9-dimensional Catalan numbers.

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A361190 Number of 4n-step lattice paths starting and ending at (0,0) that do not go above the diagonal x=y or below the x-axis using steps in {(1,1), (1,-1), (-1,0)}.

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A065077 Triangle read by rows: T(n,m) = C[n,m,m] where C[i,j,k] is the 3-dimensional Catalan pyramid defined by C[0,0,0]=1 and C[i,j,k]=0 if j>i or k>j and C[i,j,k]=C[i-1,j,k]+C[i,j-1,k]+C[i,j,k-1].

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A087727 Triangle read by rows: T(n,k) = (2 * (binomial(n,k)) * (n + 2 * k + 3)!)/((k + 1)! * (k + 2)! * (n + 3)!).

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PARI

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A123691 a(n) = number of standard Young tableaux of type (n,n-1,n-1).

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Maple

Mathematica

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A364439 a(n) is the number of paths with length 3*n that begin at (0,0), end at (0,0), and do not reach (0,0) at any point in between while 0 <= y <= x at every step, where a path is a sequence of steps in the form (1,1), (1,-1), and (-2,0).

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