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A256644 Numbers of alternating permutations where numbers at odd positions and even positions are monotone respectively. 1
1, 1, 1, 2, 5, 6, 9, 12, 21, 30, 58, 86, 176, 266, 563, 860, 1861, 2862, 6294, 9726, 21660, 33594, 75584, 117574, 266800, 416026, 950914, 1485802, 3417342, 5348882, 12369287, 19389692, 45052517, 70715342, 165002462, 259289582, 607283492, 955277402, 2244901892 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Ran Pan, Exercise Q, Project P

FORMULA

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.

EXAMPLE

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).

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).

MAPLE

C:= n-> binomial(2*n, n)/(n+1):

a:= n-> `if`(n<4, [1$3, 2][n+1], C(iquo(n, 2))+C(iquo(n-1, 2))+2):

seq(a(n), n=0..40);  # Alois P. Heinz, Apr 08 2015

MATHEMATICA

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 *)

PROG

(PARI) C(n) = binomial(2*n, n)/(n+1);

a(n) = if (n<3, 1, if (n==3, 2, C(n\2)+ C((n-1)\2)+2)); \\ Michel Marcus, Apr 07 2015

(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

(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

CROSSREFS

Cf. A104722, A000111, A000108.

Sequence in context: A161910 A151920 A160714 * A219764 A286003 A094350

Adjacent sequences:  A256641 A256642 A256643 * A256645 A256646 A256647

KEYWORD

nonn

AUTHOR

Ran Pan, Apr 07 2015

STATUS

approved

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Last modified September 26 06:54 EDT 2017. Contains 292502 sequences.