cp's OEIS Frontend

A238803 Number of ballot sequences of length 2n with exactly n fixed points.

%I A238803 #32 Sep 04 2019 16:16:11
%S A238803 1,1,3,9,29,99,357,1351,5343,21993,93923,414969,1892277,8887291,
%T A238803 42912261,212676951,1080355463,5617772049,29868493827,162204146857,
%U A238803 898874710797,5078665886931,29232738375653,171294038649639,1021117638212079,6188701520663929
%N A238803 Number of ballot sequences of length 2n with exactly n fixed points.
%C A238803 The fixed points are in the positions 1,2,...,n.
%C A238803 Also the number of standard Young tableaux with 2n cells where n is the length of the maximal consecutive sequence 1,2,...,k in the first column.  An alternate definition uses the first row.
%C A238803 All terms are odd because the counted structures come in pairs with exactly one exception.
%C A238803 Except for a(0), first differences of A005425. - _Ivan N. Ianakiev_, Sep 01 2019
%H A238803 Joerg Arndt and Alois P. Heinz, <a href="/A238803/b238803.txt">Table of n, a(n) for n = 0..750</a>
%H A238803 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F A238803 a(n) = ((2*n-1)*a(n-1)+n*(n-2)*a(n-2))/(n-1) for n>1, a(0) = a(1) = 1.
%F A238803 a(n) = A238802(2*n,n).
%F A238803 a(n) = Sum_{k=0..n} C(n-1,k) * A000085(n-k).
%F A238803 a(n) ~ exp(2*sqrt(n)-n/2-1) * n^(n/2) / sqrt(2) * (1 - 1/(6*sqrt(n))). - _Vaclav Kotesovec_, Mar 07 2014
%e A238803 For n=3 we have the following a(3) = 9 ballot sequences: [1,2,3,1,1,1], [1,2,3,1,2,3], [1,2,3,1,1,2], [1,2,3,1,2,1], [1,2,3,1,4,1], [1,2,3,1,4,2], [1,2,3,1,1,4], [1,2,3,1,2,4], [1,2,3,1,4,5].
%e A238803 Their corresponding tableaux are:
%e A238803 : 1456  14 : 145  146 : 146  14 : 145  14 : 14 :
%e A238803 : 2     25 : 26   25  : 2    26 : 2    25 : 2  :
%e A238803 : 3     36 : 3    3   : 3    3  : 3    3  : 3  :
%e A238803 :          :          : 5    5  : 6    6  : 5  :
%e A238803 :          :          :         :         : 6  :
%p A238803 a:= proc(n) option remember; `if`(n<2, 1,
%p A238803       ((2*n-1) *a(n-1) +n*(n-2) *a(n-2)) / (n-1))
%p A238803     end:
%p A238803 seq(a(n), n=0..35);
%t A238803 RecurrenceTable[{a[0]==a[1]==1,a[n]==((2n-1)a[n-1]+n(n-2)a[n-2])/(n-1)},a,{n,30}] (* _Harvey P. Dale_, Jun 25 2014 *)
%Y A238803 Cf. A000085, A005425, A238802.
%K A238803 nonn,easy
%O A238803 0,3
%A A238803 _Joerg Arndt_ and _Alois P. Heinz_, Mar 05 2014