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