cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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

Original entry on oeis.org

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

Views

Author

Peter Luschny, Sep 10 2021

Keywords

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).

Crossrefs

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

Programs

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

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).

A346719 a(n) is the number of positive Euler permutations of order 2*n. Bisection (even indices) of A347601.

Original entry on oeis.org

1, 0, 7, 102, 8109, 642220, 89458803, 15935870034, 3858227881945, 1176448390679256, 447692501190569823, 206713705368363820990, 114132862919751113790597, 74179275137980421348697732, 56081703047542413155379531979, 48790316146471264354636437276330, 48400301382766335524903922737193393
Offset: 0

Views

Author

Peter Luschny, Sep 09 2021

Keywords

Comments

For definitions and comments see A347601.

Crossrefs

Programs

  • Maple
    A346719 := n -> (A000166(2*n) + euler(2*n)) / 2:
    seq(A346719(n), n = 0..16);
  • Mathematica
    A346719[n_] := Subfactorial[2 n]/2 + Im[PolyLog[-2 n, I]];
    Table[A346719[n], {n, 0, 16}]

Formula

a(n) + A346720(n) = A000166(2n) (rencontres numbers).
a(n) - A346720(n) = A000364(n) (Euler secant numbers).
a(n) = subfactorial(2*n) / 2 + Im(PolyLog(-2*n, i)).
Showing 1-2 of 2 results.