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COMMENTS
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This is an autosequence of the first kind (array of successive differences shows typical zero diagonal).
Last digits are apparently of period 20.
From A226158(n) for the continuity of autosequences of the first kind.
b(n) = 0, 1, -1, 0, 1, 0, -3, 0, 17, ... = A226158(n) with 1 as second term instead of -1.
c(n) = 0, 0, -1, 0, 1, 0, -3, 0, 17, ... = A226158(n) with 0 as second term instead of -1.
Respective difference tables:
0, -1, -1, 0, 1, 0, -3, 0, 17, ...
-1, 0, 1, 1, -1, -3 , 3, 17, -17, ...
1, 1, 0, -2, -2, 6, 14, -34, -138, ...
etc,
0, 1, -1, 0, 1, 0, -3, 0, 17, ... = 0 followed by A036968(n+1)
1, -2, 1, 1, -1, -3, 3, 17, -17, ...
-3, 3, 0, -2, -2, 6, 14, -34, -138, ...
etc,
0, 0, -1, 0, 1, 0, -3, 0, 17, ...
0, -1, 1, 1, -1, -3, 3, 17, -17, ...
-1, 2, 0, -2, -2, 6, 14, -34, -138, ...
etc.
Since it is in the three tables, a(n) is the core of the Genocchi numbers.
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MATHEMATICA
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g[0] = 0; g[1] = -1; g[n_] := n*EulerE[n-1, 0]; G = Table[g[n], {n, 0, 30}]; Drop[Differences[G, 2], 2]
(* or, from Seidel's triangle A014781: *)
max = 26; T[1, 1] = 1; T[n_, k_] /; 1 <= k <= (n + 1)/2 := T[n, k] = If[EvenQ[n], Sum[T[n - 1, i], {i, k, max}], Sum[T[n - 1, i], {i, 1, k}]]; T[_, _] = 0; a[n_] := With[{k = Floor[(n - 1)/2] + 1}, (-1)^k*T[n + 3, k]]; Table[a[n], {n, 0, max}]
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