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A208616 Number of Young tableaux with 3 n-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing). 2
1, 1, 10, 53, 491, 6091, 87781, 1386529, 23374495, 414325055, 7646034683, 145862292213, 2861143072425, 57468095412921, 1178095930854841, 24584089994286121, 521086299342539671, 11198784502153759831, 243661974373753909051, 5360563436205104422681 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

FORMULA

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

MAPLE

a:= proc(n) option remember; `if`(n<5, [1, 1, 10, 53, 491][n+1],

      ((116013096898*n^6 -1106227006064*n^5 +3651730072724*n^4

      -5019246600372*n^3 +2923780805838*n^2 -701199942904*n) *a(n-1)

      +(-429126244301*n^6 +4283495440027*n^5 -14793057372915*n^4

      +19089754215809*n^3 -168467698444*n^2 -17547244920336*n

      +9564646580160) *a(n-2) +(24700698282*n^6 +2323122442728*n^5

      -31157649402714*n^4 +153639646198428*n^3 -363480023453028*n^2

      +415894667210784*n -184360926114960) *a(n-3) +(292122384552*n^6

      -5522876986500*n^5 +42303228071580*n^4 -167574646102140*n^3

      +360649174254588*n^2 -397826818736400*n +174796279534800) *a(n-4))/

      (n*(3709935431*n^5 -22486109809*n^4 +4251368675*n^3 +135507711725*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-Fran├žois Alcover, Mar 29 2017, after Alois P. Heinz (cf. A208615) *)

CROSSREFS

Row n=3 of A208615.

Sequence in context: A301349 A007035 A093187 * A219169 A292058 A152762

Adjacent sequences:  A208613 A208614 A208615 * A208617 A208618 A208619

KEYWORD

nonn,walk

AUTHOR

Alois P. Heinz, Feb 29 2012

STATUS

approved

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Last modified May 28 09:09 EDT 2022. Contains 354112 sequences. (Running on oeis4.)