A208616 Number of Young tableaux with 3 n-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing).

1, 1, 10, 53, 491, 6091, 87781, 1386529, 23374495, 414325055, 7646034683, 145862292213, 2861143072425, 57468095412921, 1178095930854841, 24584089994286121, 521086299342539671, 11198784502153759831, 243661974373753909051, 5360563436205104422681

Offset: 0

Author

Alois P. Heinz, Feb 29 2012

Keywords

• nonn
• walk

Comments

Also the number of (3*n-1)-step walks on n-dimensional cubic lattice from (1,0,...,0) to (3,3,...,3) with positive unit steps in all dimensions such that for each point (p_1,p_2,...,p_n) we have p_1<=p_2<=...<=p_n or p_1>=p_2>=...>=p_n.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..706

Crossrefs

Row n=3 of A208615.

Programs

Maple

a:= proc(n) option remember; `if`(n<5, [1, 1, 10, 53, 491][n+1],
      ((116013096898*n^6 -1106227006064*n^5 [phone]*n^4
      -5019246600372*n^3 [phone]*n^2 -701199942904*n) *a(n-1)
      +(-429126244301*n^6 [phone]*n^5 -14793057372915*n^4
      [phone]*n^3 -168467698444*n^2 -17547244920336*n
      [phone]) *a(n-2) +(24700698282*n^6 [phone]*n^5
      -31157649402714*n^4 [phone]*n^3 -363480023453028*n^2
      [phone]*n -184360926114960) *a(n-3) +(292122384552*n^6
      -5522876986500*n^5 [phone]*n^4 -167574646102140*n^3
      [phone]*n^2 -397826818736400*n [phone]) *a(n-4))/
      (n*(3709935431*n^5 -22486109809*n^4 [phone]*n^3 [phone]*n^2
      -75536091046*n -180596388856)))
    end:
seq(a(n), n=0..30);

Mathematica

b[nn__] := b[nn] = If[(lg = Length[{nn}]) < 2, 1, If[First[{nn}] == Last[{nn}], If[First[{nn}] == 0, 1, 2*b[First[{nn}]-1, Sequence @@ Rest[{nn}]]], If[First[{nn}] > 0, b[First[{nn}] - 1, Sequence @@ Rest[{nn}]], 0] + Sum[If[{nn}[[j]] > {nn}[[j-1]], b[Sequence @@ Table[ {nn}[[i]] - If[i == j, 1, 0], {i, 1, lg}]], 0], {j, 2, lg}]]];
a[n_] := If[n == 0, 1, b[2, Sequence @@ Table[3, {n-1}]]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 29 2017, after Alois P. Heinz (cf. A208615) *)

Formula

a(n) ~ 3^(3*n+1/2) / (Pi*n^4). - Vaclav Kotesovec, Jul 16 2014