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

%I A347598 #16 Mar 30 2024 23:07:54
%S A347598 1,0,-1,2,5,-12,-61,230,1385,-6936,-50521,316682,2702765,-20359332,
%T A347598 -199360981,1754340590,19391512145,-195242324016,-2404879675441,
%U A347598 27266796955922,370371188237525,-4669829301365052,-69348874393137901,962523286888757750,15514534163557086905
%N A347598 a(n) = permanent(T(n)), where T(n) is the tangent matrix defined in A346831 and n >= 1; by convention a(0) = 1.
%C A347598 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 A347598 a(2*n) = A028296(n); a(2*n + 1) = A347597(n).
%p A347598 # Uses the function TangentMatrix from A346831.
%p A347598 A347598 := n -> `if`(n = 0, 1, LinearAlgebra:-Permanent(TangentMatrix(n))):
%p A347598 seq(A347598(n), n = 0..12);
%o A347598 (Sage)
%o A347598 def TangentMatrix(N):
%o A347598     M = matrix(N, N)
%o A347598     H = (N + 1) // 2
%o A347598     for n in range(1, N):
%o A347598         for k in range(n):
%o A347598             M[n - k - 1, k] = 1 if n < H else -1
%o A347598             M[N - n + k, N - k - 1] = -1 if n < N - H else 1
%o A347598     return M
%o A347598 def A347598(n):
%o A347598     if n == 0: return 1
%o A347598     return TangentMatrix(n).permanent()
%o A347598 print([A347598(n) for n in range(12)])
%Y A347598 Cf. A346831, A347597, A347601, A347602, A028296, A000111, A122045, A000364.
%K A347598 sign
%O A347598 0,4
%A A347598 _Peter Luschny_, Sep 12 2021