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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.

%I A072132 #27 Mar 17 2017 23:30:18
%S A072132 1,2,6,24,120,720,5040,40320,362879,3628718,39912738,478842196,
%T A072132 6221523082,87002638276,1302313974900,20763508263000,351019617373500,
%U A072132 6266271456118776,117671982989344680,2316256222907194304,47635421509263043024,1020455890785584587168
%N 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.
%H A072132 Alois P. Heinz, <a href="/A072132/b072132.txt">Table of n, a(n) for n = 1..586</a>
%H A072132 F. Bergeron and F. Gascon, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/CYT/cyt.html">Counting Young tableaux of bounded height</a>, J. Integer Sequences, Vol. 3 (2000), #00.1.7.
%H A072132 Shalosh B. Ekhad, Nathaniel Shar, and Doron Zeilberger, <a href="http://arxiv.org/abs/1504.02513">The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r</a>, arXiv:1504.02513 [math.CO], 2015.
%H A072132 Ira M. Gessel, <a href="http://dx.doi.org/10.1016/0097-3165(90)90060-A">Symmetric functions and P-recursiveness</a>, J. Combin. Theory Ser. A 53 (1990), no. 2, 257-285.
%H A072132 Nathaniel Shar, <a href="https://pdfs.semanticscholar.org/98e3/71b675789ed6ec4f9c9cd82e2dee9ca79399.pdf">Experimental methods in permutation patterns and bijective proof</a>, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.
%F A072132 a(n) ~ 1913625 * 2^(6*n + 77) / (n^(63/2) * Pi^(7/2)). - _Vaclav Kotesovec_, Sep 10 2014
%p A072132 a:= proc(n) option remember; `if`(n<4, n!,
%p A072132       (-147456*(n+4)*(n-1)^2*(n-2)^2*(n-3)^2*a(n-4)
%p A072132       +128*(33876+30709*n+6687*n^2+410*n^3)*(n-1)^2*(n-2)^2*a(n-3)
%p A072132       -4*(1092*n^5+37140*n^4+455667*n^3+2387171*n^2+4649270*n+1206000)*
%p A072132       (n-1)^2*a(n-2) +(-17075520+(22488312+(29223280+(10509820+(1764252+
%p A072132       (154164+(6804+120*n)*n)*n)*n)*n)*n)*n)*a(n-1))/
%p A072132       ((n+16)*(n+7)^2*(n+15)^2*(n+12)^2))
%p A072132     end:
%p A072132 seq(a(n), n=1..30);  # _Alois P. Heinz_, Sep 28 2012
%t A072132 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-François Alcover_, Feb 24 2016, after _Alois P. Heinz_ (A214015) *)
%Y A072132 Cf. A052399 for T_6(n), A047890 for T_5(n), A047889 for T_4(n).
%Y A072132 Column k=8 of A214015.
%K A072132 nonn
%O A072132 1,2
%A A072132 Jesse Carlsson (j.carlsson(AT)physics.unimelb.edu.au), Jun 25 2002