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A239118 Number of ballot sequences of length n with exactly 7 fixed points.

%I A239118 #13 Feb 06 2017 18:08:07
%S A239118 0,0,0,0,0,0,0,1,1,3,9,29,99,357,1351,5342,21983,93823,414198,1886424,
%T A239118 8846390,42628782,210773592,1067599984,5533263752,29307314408,
%U A239118 158484944416,874103230896,4913196556800,28120097476640,163770757573776,969858742317600
%N A239118 Number of ballot sequences of length n with exactly 7 fixed points.
%C A239118 The fixed points are in the first 7 positions.
%C A239118 Also the number of standard Young tableaux with n cells such that the first column contains 1, 2, ..., 7, but not 8.  An alternate definition uses the first row.
%H A239118 Joerg Arndt and Alois P. Heinz, <a href="/A239118/b239118.txt">Table of n, a(n) for n = 0..800</a>
%H A239118 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F A239118 See Maple program.
%F A239118 Recurrence (for n>=9): (n-8)*(n^7 - 36*n^6 + 706*n^5 - 13080*n^4 + 177169*n^3 - 1264884*n^2 + 3776364*n - 9605520)*a(n) = (n^8 - 44*n^7 + 802*n^6 - 12104*n^5 + 206449*n^4 - 2516636*n^3 + 16735788*n^2 - 68051376*n + 170709120)*a(n-1) + (n-9)*(n-7)*(n^7 - 29*n^6 + 511*n^5 - 10055*n^4 + 131224*n^3 - 805316*n^2 + 1729104*n - 6929280)*a(n-2). - _Vaclav Kotesovec_, Mar 11 2014
%F A239118 a(n) ~ sqrt(2)/11520 * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1+7/(24*sqrt(n))). - _Vaclav Kotesovec_, Mar 11 2014
%e A239118 a(7) = 1: [1,2,3,4,5,6,7].
%e A239118 a(8) = 1: [1,2,3,4,5,6,7,1].
%e A239118 a(9) = 3: [1,2,3,4,5,6,7,1,1], [1,2,3,4,5,6,7,1,2], [1,2,3,4,5,6,7,1,8].
%e A239118 a(10) = 9: [1,2,3,4,5,6,7,1,1,1], [1,2,3,4,5,6,7,1,1,2], [1,2,3,4,5,6,7,1,1,8], [1,2,3,4,5,6,7,1,2,1], [1,2,3,4,5,6,7,1,2,3], [1,2,3,4,5,6,7,1,2,8], [1,2,3,4,5,6,7,1,8,1], [1,2,3,4,5,6,7,1,8,2], [1,2,3,4,5,6,7,1,8,9].
%p A239118 b:= proc(n) option remember; `if`(n<4, [1, 1, 3, 9][n+1],
%p A239118       ((41*n^2 +82925*n -562776)*b(n-1)
%p A239118        +(174*n^3 +63287*n^2 -447840*n +352440) *b(n-2)
%p A239118        +(133*n^3 -81472*n^2 +363510*n -267096) *b(n-3)
%p A239118        -(n-4)*(30661*n^2 -93002*n -90720) *b(n-4))/
%p A239118        (174*n^2+31449*n-246768))
%p A239118     end:
%p A239118 a:=n-> `if`(n<7, 0, b(n-7)):
%p A239118 seq(a(n), n=0..40);
%t A239118 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 == 7, 1, b[n - 8, {2, 1, 1, 1, 1, 1, 1}]]; a[n_ /; n < 7] = 0; Table[ Print["a(", n, ") = ", an = a[n]]; an, {n, 0, 40}] (* _Jean-François Alcover_, Feb 06 2015, after Maple *)
%Y A239118 Column k=7 of A238802.
%K A239118 nonn,easy
%O A239118 0,10
%A A239118 _Joerg Arndt_ and _Alois P. Heinz_, Mar 10 2014