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a(n) = permanent(T(n)), where T(n) is the tangent matrix defined in A346831 and n >= 1; by convention a(0) = 1.
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%I #16 Mar 30 2024 23:07:54

%S 1,0,-1,2,5,-12,-61,230,1385,-6936,-50521,316682,2702765,-20359332,

%T -199360981,1754340590,19391512145,-195242324016,-2404879675441,

%U 27266796955922,370371188237525,-4669829301365052,-69348874393137901,962523286888757750,15514534163557086905

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

%C 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.

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

%p # Uses the function TangentMatrix from A346831.

%p A347598 := n -> `if`(n = 0, 1, LinearAlgebra:-Permanent(TangentMatrix(n))):

%p seq(A347598(n), n = 0..12);

%o (Sage)

%o def TangentMatrix(N):

%o M = matrix(N, N)

%o H = (N + 1) // 2

%o for n in range(1, N):

%o for k in range(n):

%o M[n - k - 1, k] = 1 if n < H else -1

%o M[N - n + k, N - k - 1] = -1 if n < N - H else 1

%o return M

%o def A347598(n):

%o if n == 0: return 1

%o return TangentMatrix(n).permanent()

%o print([A347598(n) for n in range(12)])

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

%K sign

%O 0,4

%A _Peter Luschny_, Sep 12 2021