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A072133 T_9(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, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628799, 39916699, 478995537, 6226736369, 87166698628, 1307240982000, 20907446718225, 355162464899601, 6384776070987990, 121061600999380138, 2413632612087046844, 50453964720806671644, 1102844526263334763556 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..561

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) ~ 30625 * 3^(4*n + 90) / (2097152 * n^40 * Pi^4). - Vaclav Kotesovec, Sep 10 2014

MAPLE

a:= proc(n) option remember;

      `if`(n<5, n!, ((-1110790863+(1520978576+(1772290401+(607308786+

       (101671498+(9464664+(500874+(14124+165*n)*n)*n)*n)*n)*n)*n)*n)*a(n-1)

       -(1129886062*n+559908333*n^2+111239576*n^3+10655238*n^4+8778*n^6

       +491700*n^5 +353895381)*(n-1)^2*a(n-2) +(258011271+234066216*n

       +58221266*n^2+5463876*n^3 +172810*n^4)*(n-1)^2*(n-2)^2*a(n-3)

       -9*(4070430+1504292*n+117469*n^2)* (n-1)^2*(n-2)^2*(n-3)^2*a(n-4)

       +893025*(n-1)^2*(n-2)^2*(n-3)^2*(n-4)^2*a(n-5)) /

       ((n+20)^2*(n+8)^2*(n+18)^2*(n+14)^2))

    end:

seq(a(n), n=0..30);  # Alois P. Heinz, Oct 10 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 <= 9, n!, g[n, 9, {}]]; 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=9 of A214015.

Sequence in context: A152710 A152705 A152702 * A230232 A319551 A232985

Adjacent sequences:  A072130 A072131 A072132 * A072134 A072135 A072136

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

a(0)=1 prepended by Alois P. Heinz, Feb 09 2017

STATUS

approved

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Last modified June 19 07:33 EDT 2019. Contains 324218 sequences. (Running on oeis4.)