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A256644 Numbers of alternating permutations where numbers at odd positions and even positions are monotone respectively.

%I A256644 #47 Sep 08 2022 08:46:11
%S A256644 1,1,1,2,5,6,9,12,21,30,58,86,176,266,563,860,1861,2862,6294,9726,
%T A256644 21660,33594,75584,117574,266800,416026,950914,1485802,3417342,
%U A256644 5348882,12369287,19389692,45052517,70715342,165002462,259289582,607283492,955277402,2244901892
%N A256644 Numbers of alternating permutations where numbers at odd positions and even positions are monotone respectively.
%H A256644 Alois P. Heinz, <a href="/A256644/b256644.txt">Table of n, a(n) for n = 0..1000</a>
%H A256644 Ran Pan, <a href="http://www.math.ucsd.edu/~projectp/warmups/eQ.html">Exercise Q</a>, Project P
%F A256644 For n>3, a(n) = C(floor(n/2))+ C(floor((n-1)/2))+2, where C(n) is the n-th Catalan number, with a(0)=a(1)=a(2)=1 and a(3)=2.
%e A256644 a(5) = 6: (1,3,2,5,4), (1,4,2,5,3), (1,5,2,4,3), (3,4,2,5,1), (3,5,2,4,1), (4,5,2,3,1).
%e A256644 a(6) = 9: (1,3,2,5,4,6), (1,4,2,5,3,6), (1,6,2,5,3,4), (3,4,2,5,1,6), (3,6,2,5,1,4), (4,6,2,5,1,3), (4,6,3,5,1,2), (5,6,2,4,1,3), (5,6,3,4,1,2).
%p A256644 C:= n-> binomial(2*n, n)/(n+1):
%p A256644 a:= n-> `if`(n<4, [1$3, 2][n+1], C(iquo(n, 2))+C(iquo(n-1, 2))+2):
%p A256644 seq(a(n), n=0..40);  # _Alois P. Heinz_, Apr 08 2015
%t A256644 Table[Which[n < 3, 1, n == 3, 2, True, CatalanNumber[Floor[n/2]] + CatalanNumber[Floor[(n - 1)/2]] + 2], {n, 0, 38}] (* _Michael De Vlieger_, Apr 07 2015 *)
%o A256644 (PARI) C(n) = binomial(2*n, n)/(n+1);
%o A256644 a(n) = if (n<3, 1, if (n==3, 2, C(n\2)+ C((n-1)\2)+2)); \\ _Michel Marcus_, Apr 07 2015
%o A256644 (PARI) a(n) = if (n<4, return(max(1,n-1))); binomial(n\2*2, n\2)/(n\2+1)*if(n%2, 2, (5*n-2)/(4*n-4)) + 2 \\ _Charles R Greathouse IV_, Apr 07 2015
%o A256644 (Magma) [1,1,1,2] cat [Catalan(Floor(n/2))+ Catalan(Floor((n-1)/2))+2: n in [4..40]]; // _Vincenzo Librandi_, Apr 08 2015
%Y A256644 Cf. A104722, A000111, A000108.
%K A256644 nonn
%O A256644 0,4
%A A256644 _Ran Pan_, Apr 07 2015