A362603 - OEIS

A362603

Number of permutations p of [2n] in which exactly the first n terms satisfy the up-down property p(1) < p(2) > p(3) < ... .

1, 1, 4, 90, 3024, 176400, 14731200, 1710268560, 261131270400, 50881298307840, 12308045700787200, 3620112665116147200, 1272148028456410828800, 526419950201914728960000, 253357552054376603817984000, 140324455080520735061157120000, 88618646911930055808757309440000

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OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..234

FORMULA

a(n) = (2*n)!/n! * (A000111(n) - A000111(n+1)/(n+1)) for n > 0, a(0) = 1.

a(n) = A092580(2n,n).

EXAMPLE

a(0) = 1: (), the empty permutation.

a(1) = 1: 21.

a(2) = 4: 1234, 1243, 1342, 2341.

a(3) = 90: 143256, 143265, 153246, 153264, ..., 564213, 564231, 564312, 564321.

a(4) = 3024: 13245678, 13245687, 13245768, 13245786, ..., 78456213, 78456231, 78456312, 78456321.

MAPLE

b:= proc(u, o) option remember; `if`(u+o=0, 1,

add(b(o-1+j, u-j), j=1..u))

end:

a:= n-> (2*n)!/n!*`if`(n=0, 1, b(n, 0)-b(n+1, 0)/(n+1)):

seq(a(n), n=0..19);

PROG

(Python)

from fractions import Fraction

from math import factorial

from itertools import count, islice, accumulate

def A362603_gen(): # generator of terms

yield 1

blist, c = (0, 1), 1

for n in count(1):

blist, a, c = tuple(accumulate(reversed(blist), initial=0)), blist[-1], c*((n<<2)-2)

yield int(c*(a-Fraction(blist[-1], (n+1))))

A362603_list = list(islice(A362603_gen(), 20)) # Chai Wah Wu, Apr 28 2023

CROSSREFS

Cf. A000111, A001813, A092580.

Sequence in context: A067767 A240379 A304190 * A274056 A339987 A202918

Adjacent sequences: A362600 A362601 A362602 * A362604 A362605 A362606

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Apr 27 2023

STATUS

approved