%I #12 Feb 06 2017 18:07:35

%S 0,0,0,0,0,0,1,1,3,9,29,99,357,1350,5334,21912,93352,410988,1866492,

%T 8720924,41866020,206085480,1039120104,5358418224,28235017104,

%U 151838491408,832730775888,4652886489840,26470731088016,153207256585824,901628675631456

%N Number of ballot sequences of length n with exactly 6 fixed points.

%C The fixed points are in the first 6 positions.

%C Also the number of standard Young tableaux with n cells such that the first column contains 1, 2, ..., 6, but not 7. An alternate definition uses the first row.

%H Joerg Arndt and Alois P. Heinz, <a href="/A239117/b239117.txt">Table of n, a(n) for n = 0..800</a>

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

%F See Maple program.

%F Recurrence (for n>=8): (n-7)*(n^6 - 28*n^5 + 350*n^4 - 3850*n^3 + 37569*n^2 - 201082*n + 556800)*a(n) = (n^7 - 35*n^6 + 441*n^5 - 3220*n^4 + 31444*n^3 - 314265*n^2 + 1921954*n - 5066880)*a(n-1) + (n-8)*(n-6)*(n^6 - 22*n^5 + 225*n^4 - 2710*n^3 + 27854*n^2 - 136228*n + 389760)*a(n-2). - _Vaclav Kotesovec_, Mar 11 2014

%F a(n) ~ sqrt(2)/1680 * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1+7/(24*sqrt(n))). - _Vaclav Kotesovec_, Mar 11 2014

%e a(6) = 1: [1,2,3,4,5,6].

%e a(7) = 1: [1,2,3,4,5,6,1].

%e a(8) = 3: [1,2,3,4,5,6,1,1], [1,2,3,4,5,6,1,2], [1,2,3,4,5,6,1,7].

%e a(9) = 9: [1,2,3,4,5,6,1,1,1], [1,2,3,4,5,6,1,1,2], [1,2,3,4,5,6,1,1,7], [1,2,3,4,5,6,1,2,1], [1,2,3,4,5,6,1,2,3], [1,2,3,4,5,6,1,2,7], [1,2,3,4,5,6,1,7,1], [1,2,3,4,5,6,1,7,2], [1,2,3,4,5,6,1,7,8]].

%p b:= proc(n) option remember; `if`(n<3, [1,1,3][n+1],

%p ((114*n^3 -68193*n^2 +266129*n -764878)*b(n-1)

%p +(2513*n^4 +25106*n^3 +330108*n^2 -382379*n +208440)*b(n-2)

%p +(n-3)*(2399*n^3 +95128*n^2 +269793*n +65880)*b(n-3))/

%p (2513*n^3+10142*n^2+201063*n-630958))

%p end:

%p a:=n-> `if`(n<6, 0, b(n-6)):

%p seq(a(n), n=0..40);

%t b[n_, l_List] := b[n, l] = If[n <= 0, 1, b[n - 1, Append[l, 1]] + Sum[If[i == 1 || l[[i - 1]] > l[[i]], b[n - 1, ReplacePart[l, i -> l[[i]] + 1]], 0], {i, 1, Length[l]}]]; a[n_] := If[n == 6, 1, b[n - 7, {2, 1, 1, 1, 1, 1}]]; a[n_ /; n < 6] = 0; Table[ Print["a(", n, ") = ", an = a[n]]; an, {n, 0, 40}] (* _Jean-François Alcover_, Feb 06 2015, after Maple *)

%Y Column k=6 of A238802.

%K nonn,easy

%O 0,9

%A _Joerg Arndt_ and _Alois P. Heinz_, Mar 10 2014