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A339058 a(n) = 4^n*Euler(n, 1/4)*2^(valuation_{2}(n + 1)).

%I A339058 #17 Mar 16 2022 02:49:37
%S A339058 1,-2,-3,44,57,-722,-2763,196888,250737,-5746082,-36581523,2049374444,
%T A339058 7828053417,-259141449842,-2309644635483,705775346640176,
%U A339058 898621108880097,-38901437271432002,-445777636063460643,43136210244502819244,274613643571568682777,-14685255919931552812562
%N A339058 a(n) = 4^n*Euler(n, 1/4)*2^(valuation_{2}(n + 1)).
%H A339058 Michael De Vlieger, <a href="/A339058/b339058.txt">Table of n, a(n) for n = 0..432</a>
%e A339058 The array of the general case starts:
%e A339058 [k]
%e A339058 [1] 1,  1,  0, -1,   0,     1,     0,     -17,       0, ... [A198631]
%e A339058 [2] 1,  0, -1,  0,   5,     0,   -61,       0,    1385, ... [A122045]
%e A339058 [3] 1, -1, -2, 13,  22,  -121,  -602,   18581,   30742, ... [A156179]
%e A339058 [4] 1, -2, -3, 44,  57,  -722, -2763,  196888,  250737, ... [this sequence]
%e A339058 [5] 1, -3, -4, 99, 116, -2523, -8764, 1074243, 1242356, ... [A156182]
%e A339058 ...
%p A339058 a := n -> 4^n*euler(n, 1/4)*2^padic[ordp](n+1, 2): seq(a(n), n=0..9);
%t A339058 Array[4^#*EulerE[#, 1/4]*2^IntegerExponent[# + 1, 2] &, 22, 0] (* _Michael De Vlieger_, Mar 15 2022 *)
%o A339058 (SageMath)
%o A339058 def euler_sum(n):
%o A339058     return (-1)^n*sum(2^k*binomial(n, k)*euler_number(k) for k in (0..n))
%o A339058 def a(n): return euler_sum(n) << valuation(n + 1, 2)
%o A339058 print([a(n) for n in range(22)])
%Y A339058 Cf. A198631, A122045, A156179, A156182, A156191, A339057.
%Y A339058 Note the difference from A001586, A188458, and A212435.
%K A339058 sign
%O A339058 0,2
%A A339058 _Peter Luschny_, Nov 27 2020