cp's OEIS Frontend

A145870 Number of involutions of length 2n which are invariant under the reverse-complement map and have no decreasing subsequences of length 8.

%I A145870 #18 Feb 02 2025 09:35:53
%S A145870 1,2,6,20,75,301,1287,5762,26875,129520,642452,3264834,16950089,
%T A145870 89646090,482012650,2629809994,14537429823,81313943942,459705628930,
%U A145870 2624247237560,15113949789357,87755911422989,513357330465591,3023830805847910,17925386942479025
%N A145870 Number of involutions of length 2n which are invariant under the reverse-complement map and have no decreasing subsequences of length 8.
%C A145870 a(n) is also the number of involutions of length 2n+1 which are invariant under the reverse-complement map and have no decreasing subsequences of length 8.
%F A145870 a(n) = Sum_{j=0..n} C(n,j) * A000108(floor((j+1)/2)) * A000108(ceiling((j+1)/2)) * A001006(n-j), where C(n,j) = n!/(j!*(n-j)!), A000108(n) = Catalan(n) and A001006(n) = Motzkin(n).
%F A145870 From _Vaclav Kotesovec_, Feb 18 2015: (Start)
%F A145870 Recurrence: (n+3)*(n+5)*(n+6)*(192*n^2 + 992*n + 1321)*a(n) = 4*(192*n^5 + 3392*n^4 + 21897*n^3 + 64596*n^2 + 84418*n + 35925)*a(n-1) + 2*(n-1)*(3264*n^4 + 28000*n^3 + 74185*n^2 + 47329*n - 41250)*a(n-2) - 4*(n-2)*(n-1)*(3648*n^3 + 30272*n^2 + 73819*n + 38895)*a(n-3) - 105*(n-3)*(n-2)*(n-1)*(192*n^2 + 1376*n + 2505)*a(n-4).
%F A145870 a(n) ~ 7^(n+9/2) / (4 * Pi^(3/2) * n^(9/2)). (End)
%t A145870 Array[Cat, 21, 0];
%t A145870 For[i = 0, i < 21, ++i, Cat[i] = (1/(i + 1))*Binomial[2*i, i]];
%t A145870 Array[Mot, 21, 0];
%t A145870 For[i = 0, i < 21, ++i, Mot[i] = Sum[Binomial[i, 2*j]*Cat[j], {j, 0, Floor[i/2]}]];
%t A145870 Table[Sum[ Binomial[n, j]*Cat[Floor[(j + 1)/2]]*Cat[Ceiling[(j + 1)/2]]* Mot[n - j],
%t A145870 {j, 0, n}], {n, 0, 15}]
%Y A145870 Cf. A000108, A001006.
%K A145870 nonn
%O A145870 0,2
%A A145870 _Eric S. Egge_, Oct 22 2008
%E A145870 More terms from _Vaclav Kotesovec_, Feb 18 2015