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Number of standard Young tableaux with n cells and 3 as last value in the first row.
2

%I #14 Feb 06 2017 18:13:43

%S 2,3,5,10,19,41,86,197,449,1087,2650,6722,17227,45267,120069,323442,

%T 877777,2405399,6628760,18384040,51204735,143252991,402115301,

%U 1132464571,3197928097,9053803101,25689876776,73047889402,208100836969,593897902349,1697686011406

%N Number of standard Young tableaux with n cells and 3 as last value in the first row.

%C Also the number of ballot sequences of length n where 3 is the position of the last occurrence of the minimal value.

%H Joerg Arndt and Alois P. Heinz, <a href="/A245001/b245001.txt">Table of n, a(n) for n = 3..1000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>

%F Recurrence: see Maple program.

%F a(n) ~ 3^(n-3/2) / (2*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jul 11 2014

%e a(4) = 3:

%e [1 3] [1 3] [1 2 3]

%e [2] [2 4] [4]

%e [4]

%p a:= proc(n) option remember; `if`(n<5, [0$3, 2, 3][n+1],

%p ((n-2)*(30*n^4-505*n^3+3108*n^2-8147*n+7338)*a(n-1)

%p +(18703*n^3-76648*n^2+154520*n-122616-2240*n^4+105*n^5)*a(n-2)

%p -2*(n-5)*(60*n^4-965*n^3+5766*n^2-15082*n+14364)*a(n-3)

%p -12*(n-5)*(n-6)*(15*n^3-185*n^2+744*n-994)*a(n-4)) /

%p ((n-1)*(n-2)*(15*n^3-230*n^2+1159*n-1938)))

%p end:

%p seq(a(n), n=3..40);

%t b[n_, l_List] := b[n, l] = If[n == 0, 1, Sum[If[i == 1 || l[[i-1]] > l[[i]], b[n-1, ReplacePart[l, i -> l[[i]]+1]], 0], {i, 1, Length[l]}] + Function[{p}, p+(x^(1 + Total[l])-1)*Coefficient[p, x, 0]][b[n-1, Append[l, 1]]]]; a[n_] := Coefficient[ b[n, {}], x, 3]; Table[Print["a(", n, ") = ", an = a[n]]; an, {n, 3, 40}] (* _Jean-François Alcover_, Feb 06 2015, after Maple code in A238794 *)

%Y Column k=3 of A238794.

%K nonn

%O 3,1

%A _Joerg Arndt_ and _Alois P. Heinz_, Jul 09 2014