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A291982 a(n) = Euler(n, n+1) * 2^valuation(n+1, 2), where Euler(n, x) denotes the Euler polynomial.

%I A291982 #16 Jul 08 2022 08:22:55
%S A291982 1,3,6,161,380,9251,68922,9718545,24721272,1140755269,14712346550,
%T A291982 1678097074579,13104139232340,889926827467887,16319429252249970,
%U A291982 10286621696853755681,27076409740571217392,2427916115944458451025,57728302956904672126062
%N A291982 a(n) = Euler(n, n+1) * 2^valuation(n+1, 2), where Euler(n, x) denotes the Euler polynomial.
%C A291982 Conjecture: If n >= 2 is even then n*(n+1) divides a(n).
%C A291982 This conjecture was inspired by _Vladimir Shevelev_'s conjecture in A291897.
%F A291982 a(n) = Euler(n, n+1)*2^A007814(n+1).
%p A291982 A291982 := n -> euler(n, n+1)*2^(padic[ordp](n+1, 2)):
%p A291982 seq(A291982(n), n=0..18);
%t A291982 Table[2^IntegerExponent[n+1, 2] EulerE[n, n+1], {n, 1, 15}]
%o A291982 (Python)
%o A291982 from sympy import euler
%o A291982 def A291982(n): return euler(n,n+1)*(n+1 & -n-1) # _Chai Wah Wu_, Jul 07 2022
%Y A291982 Cf. A007814, A291897.
%K A291982 nonn
%O A291982 0,2
%A A291982 _Peter Luschny_, Sep 22 2017