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A072132 T_8(n) in the notation of Bergeron et al., u_k(n) in the notation of Gessel: Related to Young tableaux of bounded height. 3
1, 2, 6, 24, 120, 720, 5040, 40320, 362879, 3628718, 39912738, 478842196, 6221523082, 87002638276, 1302313974900, 20763508263000, 351019617373500, 6266271456118776, 117671982989344680, 2316256222907194304, 47635421509263043024, 1020455890785584587168 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..586

F. Bergeron and F. Gascon, Counting Young tableaux of bounded height, J. Integer Sequences, Vol. 3 (2000), #00.1.7.

Shalosh B. Ekhad, Nathaniel Shar, and Doron Zeilberger, The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r, arXiv:1504.02513 [math.CO], 2015.

Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), no. 2, 257-285.

Nathaniel Shar, Experimental methods in permutation patterns and bijective proof, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.

FORMULA

a(n) ~ 1913625 * 2^(6*n + 77) / (n^(63/2) * Pi^(7/2)). - Vaclav Kotesovec, Sep 10 2014

MAPLE

a:= proc(n) option remember; `if`(n<4, n!,

      (-147456*(n+4)*(n-1)^2*(n-2)^2*(n-3)^2*a(n-4)

      +128*(33876+30709*n+6687*n^2+410*n^3)*(n-1)^2*(n-2)^2*a(n-3)

      -4*(1092*n^5+37140*n^4+455667*n^3+2387171*n^2+4649270*n+1206000)*

      (n-1)^2*a(n-2) +(-17075520+(22488312+(29223280+(10509820+(1764252+

      (154164+(6804+120*n)*n)*n)*n)*n)*n)*n)*a(n-1))/

      ((n+16)*(n+7)^2*(n+15)^2*(n+12)^2))

    end:

seq(a(n), n=1..30);  # Alois P. Heinz, Sep 28 2012

MATHEMATICA

h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}] ]; g[n_, i_, l_] := If[n==0 || i==1, h[Join[l, Array[1&, n]]]^2, If[i < 1, 0, Sum[g[n - i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; a[n_] := If[n <= 8, n!, g[n, 8, {}]]; Table[a[n], {n, 1, 30}] (* Jean-Fran├žois Alcover, Feb 24 2016, after Alois P. Heinz (A214015) *)

CROSSREFS

Cf. A052399 for T_6(n), A047890 for T_5(n), A047889 for T_4(n).

Column k=8 of A214015.

Sequence in context: A193936 A226440 A248841 * A230231 A066459 A269221

Adjacent sequences:  A072129 A072130 A072131 * A072133 A072134 A072135

KEYWORD

nonn

AUTHOR

Jesse Carlsson (j.carlsson(AT)physics.unimelb.edu.au), Jun 25 2002

STATUS

approved

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Last modified May 21 08:53 EDT 2019. Contains 323441 sequences. (Running on oeis4.)