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%I #28 Mar 29 2017 09:55:21
%S 1,1,10,53,491,6091,87781,1386529,23374495,414325055,7646034683,
%T 145862292213,2861143072425,57468095412921,1178095930854841,
%U 24584089994286121,521086299342539671,11198784502153759831,243661974373753909051,5360563436205104422681
%N Number of Young tableaux with 3 n-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing).
%C 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.
%H Alois P. Heinz, <a href="/A208616/b208616.txt">Table of n, a(n) for n = 0..706</a>
%F a(n) ~ 3^(3*n+1/2) / (Pi*n^4). - _Vaclav Kotesovec_, Jul 16 2014
%p a:= proc(n) option remember; `if`(n<5, [1, 1, 10, 53, 491][n+1],
%p ((116013096898*n^6 -1106227006064*n^5 +3651730072724*n^4
%p -5019246600372*n^3 +2923780805838*n^2 -701199942904*n) *a(n-1)
%p +(-429126244301*n^6 +4283495440027*n^5 -14793057372915*n^4
%p +19089754215809*n^3 -168467698444*n^2 -17547244920336*n
%p +9564646580160) *a(n-2) +(24700698282*n^6 +2323122442728*n^5
%p -31157649402714*n^4 +153639646198428*n^3 -363480023453028*n^2
%p +415894667210784*n -184360926114960) *a(n-3) +(292122384552*n^6
%p -5522876986500*n^5 +42303228071580*n^4 -167574646102140*n^3
%p +360649174254588*n^2 -397826818736400*n +174796279534800) *a(n-4))/
%p (n*(3709935431*n^5 -22486109809*n^4 +4251368675*n^3 +135507711725*n^2
%p -75536091046*n -180596388856)))
%p end:
%p seq(a(n), n=0..30);
%t 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}]]];
%t a[n_] := If[n == 0, 1, b[2, Sequence @@ Table[3, {n-1}]]];
%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 29 2017, after _Alois P. Heinz_ (cf. A208615) *)
%Y Row n=3 of A208615.
%K nonn,walk
%O 0,3
%A _Alois P. Heinz_, Feb 29 2012
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