A214722 -id:A214722 - cp's OEIS frontend

A006335 a(n) = 4^n(3n)!/((n+1)!(2n+1)!).

1, 2, 16, 192, 2816, 46592, 835584, 15876096, 315031552, 6466437120, 136383037440, 2941129850880, 64614360416256, 1442028424527872, 32619677465182208, 746569714888605696, 17262927525017812992, 402801642250415636480, 9474719710174783733760, 224477974671833337692160

Offset: 0

N. J. A. Sloane

Number of planar lattice walks of length 3n starting and ending at (0,0), remaining in the first quadrant and using only NE,W,S steps.
Equals row sums of triangle A140136. - Michel Marcus, Nov 16 2014
Number of linear extensions of the poset V x [n], where V is the 3-element poset with one least element and two incomparable elements: see Kreweras and Niederhausen (1981) and Hopkins and Rubey (2020) references. - Noam Zeilberger, May 28 2020

			G.f. = 1 + 2*x + 16*x^2 + 192*x^3 + 2816*x^4+ 46592*x^5 + 835584*x^6 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..700
Andrei Asinowski, Cyril Banderier, and Sarah J. Selkirk, From Kreweras to Gessel: A walk through patterns in the quarter plane, Séminaire Lotharingien de Combinatoire, Proc. 35th Conf. Formal Power Series and Alg. Comb. (Davis, 2023) Vol. 89B, Art. #30.
Olivier Bernardi, Bijective counting of Kreweras walks and loopless triangulations, Journal of Combinatorial Theory, Series A 114:5 (2007), 931-956.
M. Bousquet-Mélou, Walks in the quarter plane: Kreweras' algebraic model, arXiv:math/0401067 [math.CO], 2004-2006.
M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008.
Sam Hopkins and Martin Rubey, Promotion of Kreweras words, arXiv:2005.14031 [math.CO], 2020.
G. Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, 6 (1965), circa p. 82.
G. Kreweras and H. Niederhausen, Solution of an enumerative problem connected with lattice paths, European J. Combin., 2 (1981), 55-60.

Equals 2^(n-1) * A000309(n-1) for n>1.
Cf. A098272. First row of array A098273.
Column of A176129, A214631, A214722, A340591.

[4^n*Factorial(3*n)/(Factorial(n+1)*Factorial(2*n+1)) : n in [0..20]]; // Wesley Ivan Hurt, Nov 16 2014

A006335:=n->4^n*(3*n)!/((n+1)!*(2*n+1)!): seq(A006335(n), n=0..20); # Wesley Ivan Hurt, Nov 16 2014

aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[aux[0, 0, 3 n], {n, 0, 25}] (* Manuel Kauers, Nov 18 2008 *)
Table[(4^n (3 n)! / ((n + 1)! (2 n + 1)!)), {n, 0, 200}] (* Vincenzo Librandi, Nov 17 2014 *)

{a(n) = if( n<0, 0, 4^n * (3*n)! / ((n+1)! * (2*n+1)!))}; /* Michael Somos, Jan 23 2003 */

def a(n):
    return (4**n * binomial(3 * n, 2 * n)) // ((n + 1) * (2 * n + 1))
# F. Chapoton, Jun 01 2020

G.f.: (1/(12*x)) * (hypergeom([ -2/3, -1/3],[1/2],27*x)-1). - Mark van Hoeij, Nov 02 2009
a(n+1) = 6*(3*n+2)*(3*n+1)*a(n)/((2+n)*(2*n+3)). - Robert Israel, Nov 17 2014
a(n) ~ 3^(3*n + 1/2) / (4*sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Mar 26 2016
E.g.f.: 2F2(1/3,2/3; 3/2,2; 27*x). - Ilya Gutkovskiy, Jan 25 2017

Edited by N. J. A. Sloane, Dec 20 2008 at the suggestion of R. J. Mathar

A213978 Number of solid standard Young tableaux of shape [[n,n,n],[n]].

Original entry on oeis.org

1, 3, 91, 5471, 464836, 48767805, 5900575762, 791402291063, 114754560003596, 17688389169462060, 2864042102057254739, 482894371222455465001, 84225614036198359288620, 15119622005825185224290830, 2782232873996840900804273236, 523114052492282720617167786279, 100231256005025286627952024093564, 19528383010645472628217323778258916

Offset: 0

N. J. A. Sloane, Jul 07 2012

nonn

Ekhad-Zeilberger give 121 terms.

Alois P. Heinz, Table of n, a(n) for n = 0..129
Shalosh B. Ekhad and Doron Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux.
S. B. Ekhad and D. Zeilberger, Number of Solid Standard Young Tableaux of shape [[n,n,n],[n]], (n=1..121); Local copy
S. B. Ekhad and D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229, 2012

Column k=3 of A214722.

b:= proc(x, y, z, u) option remember; `if`({x, y, z, u}={0}, 1,
     `if`(x>y and x>u, b(x-1, y, z, u), 0)+`if`(y>z, b(x, y-1, z, u), 0)+
     `if`(z>0, b(x, y, z-1, u), 0)+`if`(u>0, b(x, y, z, u-1), 0))
    end:
a:= n-> b(n$4):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 19 2012

b[x_, y_, z_, u_] := b[x, y, z, u] = If[Union[{x, y, z, u}] == {0}, 1, If[x>y && x>u, b[x-1, y, z, u], 0] + If[y>z, b[x, y-1, z, u], 0] + If[z>0, b[x, y, z-1, u], 0] + If[u>0, b[x, y, z, u-1], 0]]; a[n_] := b[n, n, n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 11 2014, after Alois P. Heinz *)

Conjecture: Limit n->infinity a(n)^(1/n) = 256. - Vaclav Kotesovec, Jul 17 2014

A214631 Number A(n,k) of solid standard Young tableaux of shape [[(n)^(k+1)],[n]^k]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 16, 1, 1, 20, 936, 192, 1, 1, 70, 85800, 379366, 2816, 1, 1, 252, 9962680, 1825221320, 249664758, 46592, 1, 1, 924, 1340103744, 14336196893200, 89261675900020, 221005209058, 835584, 1

Offset: 0

Alois P. Heinz, Jul 26 2012

			Square array A(n,k) begins:
  1,    1,         1,              1,                    1, ...
  1,    2,         6,             20,                   70, ...
  1,   16,       936,          85800,              9962680, ...
  1,  192,    379366,     1825221320,       14336196893200, ...
  1, 2816, 249664758, 89261675900020, 70351928759681296000, ...

Alois P. Heinz, Antidiagonals n = 0..12
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Columns k=0-2 give: A000012, A006335, A214638.
Rows n=0-1 give: A000012, A000984.
Cf. A213932, A213978, A214637, A214722.

b:= proc(l) option remember; local m; m:= nops(l);
      `if`({map(x-> x[], l)[]}={0}, 1, add(add(`if`(l[i][j]>
      `if`(i=m or nops(l[i+1])
      `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
       j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
    end:
A:= (n, k)-> b([[n$(k+1)], [n]$k]):
seq(seq(A(n, d-n), n=0..d), d=0..8);

b[l_] := b[l] = With[{m = Length[l]}, If[Union[Flatten[l]] == {0}, 1, Sum[Sum[If[l[[i, j]] > If[i == m || Length[l[[i+1]] ] < j, 0, l[[i+1, j]] ] && l[[i, j]] > If[Length[l[[i]] ] == j, 0, l[[i, j+1]] ], b[ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]]-1]]], 0], {j, 1, Length[l[[i]] ]}], {i, 1, m}]]]; a[n_, k_] := b[{Array[n&, k+1], Sequence @@ Array[{n}&, k]}]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)

A214824 Number of solid standard Young tableaux of shape [[(2)^n],[2]].

Original entry on oeis.org

2, 16, 91, 456, 2145, 9724, 43043, 187408, 806208, 3436720, 14545982, 61214960, 256411935, 1069854660, 4449173475, 18450500640, 76326664260, 315077780160, 1298203997610, 5340028714800, 21932944632690, 89963953083576, 368565304248846, 1508283816983776

Offset: 1

Alois P. Heinz, Jul 28 2012

nonn

a(n) is odd if and only if n = 3 or n in { 2^k-3, 2^k-1 : k = 3,4,5, ... }.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012.
G. Kreweras, Sur les extensions lineaires d'une famille particuliere d'ordres partiels, Discrete Math., 27 (1979), 279-295.
G. Kreweras, Sur les extensions linéaires d'une famille particulière d'ordres partiels, Discrete Math., 27 (1979), 279-295. (Annotated scanned copy)
Wikipedia, Young tableau

Row n=2 of A214722 and of A259101.

a:= proc(n) option remember; `if`(n=1, 2,
      (4+(18+(22+4*n)*n)*n)*n*a(n-1)/(6+(-13+(1+(5+n)*n)*n)*n))
    end:
seq(a(n), n=1..30);

a[1] = 2; a[n_] := a[n] = (4 + (18 + (22 + 4*n)*n)*n)*n*a[n - 1]/(6 + (-13 + (1 + (5 + n)*n)*n)*n); Array[a, 30] (* Jean-François Alcover, Nov 08 2017, translated from Maple *)

a(n) = 2*(2*n+1)*(n^2+5*n+2)*n/((n-1)*(n+3)*(n^2+3*n-2))*a(n-1); a(1) = 2.

A215220 Number of solid standard Young tableaux of shape [[n,n,n,n],[n]].

Original entry on oeis.org

1, 4, 456, 143164, 75965484, 55824699632, 51274161733160, 55418842406649988, 67819708829687672202, 91539069926354814114556, 133752944758581353219955762, 208673064320580765981337783096, 343997162091593719562479905281938, 594344377404793356460064021706935470

Offset: 0

Alois P. Heinz, Aug 06 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..40
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Column k=4 of A214722.

b:= proc(w, x, y, z, u) option remember;
     `if`({w, x, y, z, u}={0}, 1, `if`(w>x and w>u, b(w-1, x, y, z, u), 0)+
     `if`(x>y, b(w, x-1, y, z, u), 0)+ `if`(y>z, b(w, x, y-1, z, u), 0)+
     `if`(z>0, b(w, x, y, z-1, u), 0)+ `if`(u>0, b(w, x, y, z, u-1), 0))
    end:
a:= n-> b(n$5):
seq(a(n), n=0..20);

b[w_, x_, y_, z_, u_] := b[w, x, y, z, u] =
  If[Union@{w, x, y, z, u} == {0}, 1,
  If[w > x && w > u, b[w - 1, x, y, z, u], 0] +
  If[x > y, b[w, x - 1, y, z, u], 0] +
  If[y > z, b[w, x, y - 1, z, u], 0] +
  If[z > 0, b[w, x, y, z - 1, u], 0] +
  If[u > 0, b[w, x, y, z, u - 1], 0]];
a[n_] :=  b[n, n, n, n, n];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 08 2017, translated from Maple *)

A211505 Number of solid standard Young tableaux of shape [[(3)^n],[3]].

Original entry on oeis.org

5, 192, 5471, 143164, 3636776, 91442364, 2293620329, 57583680440, 1449149180310, 36577522323264, 926132803592304, 23521922685452320, 599176262762762880, 15305331537941936820, 391972832520910172865, 10062682947669846370800, 258904473727824391312650

Offset: 1

Alois P. Heinz, Aug 06 2012

nonn

a(n) is odd if and only if n = 2^k-1, k in {1, 2, 3, ... }.

Alois P. Heinz, Table of n, a(n) for n = 1..700
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Row n=3 of A214722.

Cf. A000225.

a:=proc(n) option remember; `if`(n<3, [5, 192][n],
   ((-300307980720421134*n^6 -450257143569814251*n^5 +520721206232593545*n^4
    +237675576478617990*n^3 -244917832991741721*n^2 +20947752092648421*n
    +7263907687560150)*a(n-2) +(20284627624231332*n^6 +74969736339564876*n^5
    -90577816139486502*n^4 -578021108537112633*n^3 -713543105276625459*n^2
    -309855589348004634*n -29055630750240600)*a(n-1)) / (339337400732270*n^6
    +2086412513047793*n^5 -1761594949059583*n^4 -24340927184757907*n^3
    -25005499944921313*n^2 +25705682578023740*n +29939073413286900))
   end:
seq(a(n), n=1..20);

Table[(600 + 2874*n + 4709*n^2 + 3246*n^3 + 1019*n^4 + 144*n^5 + 8*n^6)*(2+3*n)! / (2*(3+2*n)*(5+2*n)*(n-1)!*(2+n)!*(5+n)!), {n, 1, 20}] (* Vaclav Kotesovec, Jul 16 2014 *)

a(n) = (600 + 2874*n + 4709*n^2 + 3246*n^3 + 1019*n^4 + 144*n^5 + 8*n^6)*(2+3*n)! / (2*(3+2*n)*(5+2*n)*(n-1)!*(2+n)!*(5+n)!). - Vaclav Kotesovec, Jul 16 2014

A258583 Number of solid standard Young tableaux of shape [[{n}^n],[n]].

Original entry on oeis.org

1, 1, 16, 5471, 75965484, 70692556053053, 6614511157454872712100, 87353366195666890516586545068535, 217757982462900115559339884671224174403391534, 132100470099008733697710444705793312015509514686031193798241, 24431421930145927713526351934816506384811205836191828228900485331569907542066

Offset: 0

Alois P. Heinz, Nov 06 2015

nonn

S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012.
Wikipedia, Young tableau

Cf. A214722, A258586.

b:= proc(l) option remember; local m; m:= nops(l);
      `if`({map(x-> x[], l)[]}={0}, 1, add(add(`if`(l[i][j]>
      `if`(i=m or nops(l[i+1])
      `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
       j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
    end:
a:= n-> b([[n$n], [n]]):
seq(a(n), n=0..8);

b[l_] := b[l] = With[{m = Length[l]}, If[Union[Flatten[l]] == {0}, 1, Sum[Sum[ If[l[[i, j]] > If[i == m || Length[l[[i+1]]] If[Length[l[[i]]] == j, 0, l[[i, j+1]]], b[ReplacePart[l, i -> ReplacePart[ l[[i]], j -> l[[i, j]]-1]]], 0], {j, Length[l[[i]]]}], {i, m}]]];
a[n_] := b[{Array[n&, n], {n}}];
Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Aug 25 2021, after Alois P. Heinz *)

a(n) = A214722(n,n).

A258586 Number of solid standard Young tableaux of shape [[{n}^(n+1)],[n]^n].

Original entry on oeis.org

1, 2, 936, 1825221320, 70351928759681296000, 160978956785364112335731878007698260, 51488321677815455036453939239317069333712945710369620220, 4648889159675386017282064494039528050991187044317172798976485350954735075040045120

Offset: 0

Alois P. Heinz, Nov 06 2015

nonn

S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Cf. A214722, A258583.

b:= proc(l) option remember; local m; m:= nops(l);
      `if`({map(x-> x[], l)[]}={0}, 1, add(add(`if`(l[i][j]>
      `if`(i=m or nops(l[i+1])
      `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
       j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
    end:
a:= n-> b([[n$(n+1)], [n]$n]):
seq(a(n), n=0..5);

cp's OEIS Frontend

A006335 a(n) = 4^n*(3*n)!/((n+1)!*(2*n+1)!).

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A213978 Number of solid standard Young tableaux of shape [[n,n,n],[n]].

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A214631 Number A(n,k) of solid standard Young tableaux of shape [[(n)^(k+1)],[n]^k]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A214824 Number of solid standard Young tableaux of shape [[(2)^n],[2]].

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A215220 Number of solid standard Young tableaux of shape [[n,n,n,n],[n]].

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Mathematica

A211505 Number of solid standard Young tableaux of shape [[(3)^n],[3]].

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A258583 Number of solid standard Young tableaux of shape [[{n}^n],[n]].

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A006335 a(n) = 4^n(3n)!/((n+1)!(2n+1)!).