A347602 a(n) is the number of negative Euler permutations of order n.

0, 0, 1, 0, 2, 28, 163, 812, 6724, 70216, 692741, 7183944, 86756038, 1155576132, 16135231015, 239656087572, 3838836369800, 65522667301840, 1178853270354697, 22361732381344592, 447322130002332298, 9399988542176154796, 206783054242756958891, 4754731473884444589756

Offset

0,5

Comments

Let M be the tangent matrix of dimension n X n. The definition of the tangent matrix is given in A346831. An Euler permutation of order n is a permutation sigma of {1,...,n} if P = Product_{k=1..n} M(k, sigma(k)) does not vanish. We say sigma is a positive Euler permutation of order n (or sigma in EP(n)) if P = 1 and a negative Euler permutation of order n (or sigma in EN(n)) if P = -1.

a(n) = card(EN(n)), the number of negative Euler permutations of order n. A table of negative Euler permutations is given in A347767. Related sequences are A347599 (Genocchi permutations) and A347600 (Seidel permutations).

Links

Table of n, a(n) for n=0..23.

Formula

Let |S| denote the cardinality of a set S. Following identities hold for n >= 0:

A347601(n) + a(n) = |EP(n) | + |EN(n) | = A000166(n) (rencontres numbers),

A347601(2n) - a(2n) = |EP(2n)| - |EN(2n)| = A122045(n) (Euler numbers),

A347601(n) - a(n) = |EP(n) | - |EN(n) | = A347598(n).

Maple

# Uses function EulerPermutations from A347601.
A347602 := n -> `if`(n = 0, 0, EulerPermutations(n, 'neg')):
seq(A347602(n), n = 0..8);

Crossrefs

Cf. A000166, A122045, A346831, A347597, A347598, A347601 (pos. perm.), A347767 (table), A347599, A347600, A346720 (bisection even indices).

Sequence in context: A174707 A110241 A338426 * A006332 A280120 A065340

Adjacent sequences: A347599 A347600 A347601 * A347603 A347604 A347605

Keyword

nonn

Author

Peter Luschny, Sep 10 2021

Status

approved