A347598 a(n) = permanent(T(n)), where T(n) is the tangent matrix defined in A346831 and n >= 1; by convention a(0) = 1.

1, 0, -1, 2, 5, -12, -61, 230, 1385, -6936, -50521, 316682, 2702765, -20359332, -199360981, 1754340590, 19391512145, -195242324016, -2404879675441, 27266796955922, 370371188237525, -4669829301365052, -69348874393137901, 962523286888757750, 15514534163557086905

Offset

0,4

Comments

This sequence is an extension of the even-indexed Euler numbers A028296. These numbers can be extended to A000111 by adding the expansion of the tangent function, respectively considering the alternating permutations. Here one gets a different extension of the nonzero Euler numbers by considering the permutations A347601 and A347602 based on the permanent of the tangent matrix as defined in A346831. An overview gives a table in A347601.

Links

Table of n, a(n) for n=0..24.

Formula

a(2*n) = A028296(n); a(2*n + 1) = A347597(n).

Maple

# Uses the function TangentMatrix from A346831.
A347598 := n -> `if`(n = 0, 1, LinearAlgebra:-Permanent(TangentMatrix(n))):
seq(A347598(n), n = 0..12);

Prog

(Sage)
def TangentMatrix(N):
    M = matrix(N, N)
    H = (N + 1) // 2
    for n in range(1, N):
        for k in range(n):
            M[n - k - 1, k] = 1 if n < H else -1
            M[N - n + k, N - k - 1] = -1 if n < N - H else 1
    return M
def A347598(n):
    if n == 0: return 1
    return TangentMatrix(n).permanent()
print([A347598(n) for n in range(12)])

Crossrefs

Cf. A346831, A347597, A347601, A347602, A028296, A000111, A122045, A000364.

Sequence in context: A064636 A355991 A376148 * A293023 A145857 A264863

Adjacent sequences: A347595 A347596 A347597 * A347599 A347600 A347601

Keyword

sign

Author

Peter Luschny, Sep 12 2021

Status

approved