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A072167 T_10(n) in the notation of Bergeron et al., u_10(n) in the notation of Gessel: Related to Young tableaux of bounded height.

%I A072167 #23 Dec 01 2018 07:59:05
%S A072167 1,2,6,24,120,720,5040,40320,362880,3628800,39916799,479001478,
%T A072167 6227012074,87177809092,1307651456625,20921799763626,355647213494682,
%U A072167 6400805686152436,121585553747301448,2430677026538811240
%N A072167 T_10(n) in the notation of Bergeron et al., u_10(n) in the notation of Gessel: Related to Young tableaux of bounded height.
%C A072167 In general, column k > 1 of A214015 is asymptotic to (Product_{j=1..k} j!) * k^(2*n + k^2/2) / (Pi^((k-1)/2) * 2^((k-1)*(k+2)/2) * n^((k^2-1)/2)). - _Vaclav Kotesovec_, Sep 10 2014
%H A072167 Vaclav Kotesovec, <a href="/A072167/b072167.txt">Table of n, a(n) for n = 1..500</a>
%H A072167 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 A072167 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 A072167 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 A072167 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 A072167 a(n) ~ 546852789 * 2^(2*n + 26)* 5^(2*n + 55) / (n^(99/2) * Pi^(9/2)). - _Vaclav Kotesovec_, Sep 10 2014
%F A072167 Recurrence: (n+9)^2*(n + 16)^2*(n + 21)^2*(n + 24)^2*(n + 25)*a(n) = 2*(110*n^9 + 13497*n^8 + 704088*n^7 + 20286926*n^6 + 350428790*n^5 + 3672058745*n^4 + 22314826244*n^3 + 68412531600*n^2 + 64584000000*n - 45705600000)*a(n-1) - 4*(n-1)^2*(4092*n^7 + 349536*n^6 + 12084237*n^5 + 215646686*n^4 + 2088552609*n^3 + 10433770264*n^2 + 21962910556*n + 8002290720)*a(n-2) + 8*(n-2)^2*(n-1)^2*(61160*n^5 + 3273732*n^4 + 64938154*n^3 + 580006399*n^2 + 2208119423*n + 2488711560)*a(n-3) - 256*(n-3)^2*(n-2)^2*(n-1)^2*(21076*n^3 + 577824*n^2 + 4743323*n + 11018760)*a(n-4) + 7372800*(n-4)^2*(n-3)^2*(n-2)^2*(n-1)^2*(2*n + 15)*a(n-5). - _Vaclav Kotesovec_, Sep 10 2014
%p A072167 h:= proc(l) local n; n:=nops(l); add(i, i=l)! / mul(mul(1+l[i]-j
%p A072167        +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
%p A072167     end:
%p A072167 g:= proc(n, i, l) option remember;
%p A072167       `if`(n=0, h(l)^2, `if`(i<1, 0, `if`(i=1, h([l[], 1$n])^2,
%p A072167        g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i, [l[], i])))))
%p A072167     end:
%p A072167 a:= n-> g(n, 10, []):
%p A072167 seq(a(n), n=0..25); # _Vaclav Kotesovec_, Sep 10 2014, after Alois P. Heinz
%t A072167 RecurrenceTable[{-7372800*(-4 + n)^2*(-3 + n)^2*(-2 + n)^2*(-1 + n)^2*(15 + 2*n)*a[-5 + n] + 256*(-3 + n)^2*(-2 + n)^2*(-1 + n)^2*(11018760 + 4743323*n + 577824*n^2 + 21076*n^3)*a[-4 + n]-8*(-2 + n)^2*(-1 + n)^2*(2488711560 + 2208119423*n + 580006399*n^2 + 64938154*n^3 + 3273732*n^4 + 61160*n^5)*a[-3 + n] + 4*(-1 + n)^2*(8002290720 + 21962910556*n + 10433770264*n^2 + 2088552609*n^3 + 215646686*n^4 + 12084237*n^5 + 349536*n^6 + 4092*n^7)*a[-2 + n]-2*(-45705600000 + 64584000000*n + 68412531600*n^2 + 22314826244*n^3 + 3672058745*n^4 + 350428790*n^5 + 20286926*n^6 + 704088*n^7 + 13497*n^8 + 110*n^9)*a[-1 + n] + (9 + n)^2*(16 + n)^2*(21 + n)^2*(24 + n)^2*(25 + n)*a[n]==0,a[1]==1,a[2]==2,a[3]==6,a[4]==24,a[5]==120},a,{n,1,20}] (* _Vaclav Kotesovec_, Sep 10 2014 *)
%Y A072167 Cf. A052399 for T_6(n), A047890 for T_5(n), A047889 for T_4(n).
%Y A072167 Column k=10 of A214015.
%K A072167 nonn
%O A072167 1,2
%A A072167 Jesse Carlsson (j.carlsson(AT)physics.unimelb.edu.au), Jun 29 2002