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A356900 a(n) = P(n, 1/2) where P(n, x) = x^(-n)*Sum_{k=0..n} A241171(n, k)*x^k.

%I A356900 #11 Sep 03 2022 08:13:12
%S A356900 1,1,8,154,5552,321616,27325088,3200979664,494474723072,
%T A356900 97390246272256,23820397371219968,7083386168647642624,
%U A356900 2516691244849530785792,1052914814802404260765696,512347915163742179541659648,286902390859642414913802102784,183187476890368376930869730803712
%N A356900 a(n) = P(n, 1/2) where P(n, x) = x^(-n)*Sum_{k=0..n} A241171(n, k)*x^k.
%C A356900 Other special values of this Euler type polynomials are: P(n, -1) = A000364(n); P(n, -1/2) = A002105(n); P(n, 1) = A094088(n), where we always make the assumption that the offset of the sequences is 0. A partition refinement of Joffe's triangle A241171 is A327022.
%p A356900 a := n -> 2^n*add(A241171(n, k)*(1/2)^k, k = 0..n):
%p A356900 seq(a(n), n = 0..16);
%o A356900 (SageMath) # Using function PtransMatrix from A269941.
%o A356900 def E(n, v):
%o A356900     eulr = lambda n: 1 / ((2 * n - 1) * (2 * n))
%o A356900     norm = lambda n, k: (1 / v)^n * factorial(2 * n)
%o A356900     P = PtransMatrix(n, eulr, norm)
%o A356900     return [(-1)^j * sum([v^k * P[j][k] for k in range(j + 1)]) for j in range(n)]
%o A356900 A356900List = lambda n: E(n, -1/2); print(A356900List(17))
%o A356900 # A002105List = lambda n: E(n, 1/2) returns the reduced tangent numbers A002105.
%Y A356900 Cf. A241171, A327022, A000364, A002105, A094088, A269941.
%K A356900 nonn
%O A356900 0,3
%A A356900 _Peter Luschny_, Sep 03 2022