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%I #17 Feb 06 2017 17:59:43
%S 0,0,0,1,1,3,9,28,93,321,1168,4404,17328,70408,296436,1284768,5740804,
%T 26332788,124066608,598625296,2958281328,14941136784,77111251408,
%U 406028059968,2180584156176,11930067296848,66468429865344,376770132276288,2172036623279488
%N Number of ballot sequences of length n with exactly 3 fixed points.
%C The fixed points are in the first 3 positions.
%C Also the number of standard Young tableaux with n cells such that the first column contains 1, 2, and 3, but not 4. An alternate definition uses the first row.
%H Joerg Arndt and Alois P. Heinz, <a href="/A238978/b238978.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 a(n) ~ sqrt(2)/16 * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1 + 7/(24*sqrt(n))). - _Vaclav Kotesovec_, Mar 07 2014
%F Recurrence (for n>=5): (n-4)*(n^3 - 10*n^2 + 27*n - 26)*a(n) = (n^4 - 14*n^3 + 67*n^2 - 150*n + 152)*a(n-1) + (n-5)*(n-3)*(n^3 - 7*n^2 + 10*n - 8)*a(n-2). - _Vaclav Kotesovec_, Mar 08 2014
%e a(3) = 1: [1,2,3].
%e a(4) = 1: [1,2,3,1].
%e a(5) = 3: [1,2,3,1,1], [1,2,3,1,2], [1,2,3,1,4].
%e a(6) = 9: [1,2,3,1,1,1], [1,2,3,1,1,2], [1,2,3,1,1,4], [1,2,3,1,2,1], [1,2,3,1,2,3], [1,2,3,1,2,4], [1,2,3,1,4,1], [1,2,3,1,4,2], [1,2,3,1,4,5].
%p a:= proc(n) option remember; `if`(n<4, n*(n-1)*(n-2)/6,
%p ((4*n^3-54*n^2+216*n-254) *a(n-1)
%p +(n-5)*(3*n^3-31*n^2+84*n-30) *a(n-2)
%p -(n-5)*(n-6)*(n^2-3*n-8) *a(n-3)) /
%p ((n-3)*(3*n^2-33*n+86)))
%p end:
%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 == 3, 1, b[n - 4, {2, 1, 1}]]; a[n_ /; n < 3] = 0; Table[Print["a(", n, ") = ", an = a[n]]; an, {n, 0, 40}] (* _Jean-François Alcover_, Feb 06 2015, after Maple *)
%Y Column k=3 of A238802.
%K nonn,easy
%O 0,6
%A _Joerg Arndt_ and _Alois P. Heinz_, Mar 07 2014
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