OFFSET
0,2
COMMENTS
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.
LINKS
Eric Weisstein's MathWorld, Genocchi Number.
Wikipedia, Genocchi number
FORMULA
a(n) = (n+2)*E(n+1, 0) - 2*(n+3)*E(n+2, 0) + (n+4)*E(n+3, 0), where E(n,x) is the n-th Euler polynomial.
a(n) = -2*(2^(n+2)-1)*B(n+2) + 4*(2^(n+3)-1)*B(n+3) - 2*(2^(n+4)-1)*B(n+4), where B(n) is the n-th Bernoulli number.
MATHEMATICA
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}]
CROSSREFS
KEYWORD
sign
AUTHOR
Jean-François Alcover and Paul Curtz, Nov 18 2016
STATUS
approved