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a(n) = A226158(n) - 2*A226158(n+1).
4

%I #22 Jan 19 2018 19:35:24

%S 2,1,-1,-2,1,6,-3,-34,17,310,-155,-4146,2073,76454,-38227,-1859138,

%T 929569,57641238,-28820619,-2219305810,1109652905,103886563462,

%U -51943281731,-5810302084962,2905151042481,382659344967926

%N a(n) = A226158(n) - 2*A226158(n+1).

%C The array A(n,k) = A(n-1,k+1) - A(n-1,k) of the sequence in the first row and higher-order sequences in followup rows starts:

%C 2, 1, -1, -2, 1, 6, -3, ...

%C -1, -2, -1, 3, 5, -9, -31, ...

%C -1, 1, 4, 2, -14, -22, 82, ...

%C 2, 3, -2, -16, -8, 104, 160, ...

%C 1, -5, -14, 8, 112, 56, -1160, ...

%C -6, -9, 22, 104, -56, -1216, -608, ...

%C -3, 31, 82, -160, -1160, 608, 18880, ...

%C etc.

%C a(n) is an autosequence: Its inverse binomial transform is the sequence (up to a sign), which means top row and left column in the difference array have the same absolute values.

%C The main diagonal is the double of the first upper diagonal: A(n,n) = 2*A(n,n+1).

%C A(n,n+1) = (-1)^n*A005439(n), which also appears as the first upper diagonal of the difference array of A226158(n).

%H G. C. Greubel, <a href="/A230324/b230324.txt">Table of n, a(n) for n = 0..500</a>

%H OEIS Wiki, <a href="https://oeis.org/wiki/Autosequence">Autosequence</a>

%F a(n)/2 + A164555(n)/A027642(n) = 2*A225825(n)/A141056(n).

%e a(0) = 0 - 2 * (-1) = 2,

%e a(1) = -1 - 2 * (-1) = 1,

%e a(2) = -1 - 2 * 0 = -1,

%e a(3) = 0 - 2 * 1 = -2,

%e a(4) = 1 - 2 * 0 = 1,

%e a(5) = 0 - 2 * (-3) = 6.

%p A226158 := proc(n)

%p if n = 0 then

%p 0;

%p else

%p Zeta(1-n)*2*n*(2^n-1) ;

%p end if;

%p end proc:

%p A230324 := proc(n)

%p A226158(n)-2*A226158(n+1) ;

%p end proc: # _R. J. Mathar_, Oct 28 2013

%t a[0] = 2; a[1] = 1; a[n_] := n EulerE[n-1, 0] - 2 (n+1) EulerE[n, 0];

%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jun 07 2017 *)

%Y Cf. A050946.

%K sign

%O 0,1

%A _Paul Curtz_, Oct 16 2013