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%I #24 Feb 06 2023 10:01:26
%S 0,-2,-2,6,14,-34,-138,310,1918,-4146,-36154,76454,891342,-1859138,
%T -27891050,57641238,1080832286,-2219305810,-50833628826,103886563462,
%U 2853207760750,-5810302084962,-188424521441482,382659344967926,14464296482284734,-29311252309537394,-1277229462293249018
%N Shifted sequence of second differences of Genocchi numbers.
%C This is an autosequence of the first kind (array of successive differences shows typical zero diagonal).
%C Last digits are apparently of period 20.
%C From A226158(n) for the continuity of autosequences of the first kind.
%C b(n) = 0, 1, -1, 0, 1, 0, -3, 0, 17, ... = A226158(n) with 1 as second term instead of -1.
%C c(n) = 0, 0, -1, 0, 1, 0, -3, 0, 17, ... = A226158(n) with 0 as second term instead of -1.
%C Respective difference tables:
%C 0, -1, -1, 0, 1, 0, -3, 0, 17, ...
%C -1, 0, 1, 1, -1, -3 , 3, 17, -17, ...
%C 1, 1, 0, -2, -2, 6, 14, -34, -138, ...
%C etc,
%C 0, 1, -1, 0, 1, 0, -3, 0, 17, ... = 0 followed by A036968(n+1)
%C 1, -2, 1, 1, -1, -3, 3, 17, -17, ...
%C -3, 3, 0, -2, -2, 6, 14, -34, -138, ...
%C etc,
%C 0, 0, -1, 0, 1, 0, -3, 0, 17, ...
%C 0, -1, 1, 1, -1, -3, 3, 17, -17, ...
%C -1, 2, 0, -2, -2, 6, 14, -34, -138, ...
%C etc.
%C Since it is in the three tables, a(n) is the core of the Genocchi numbers.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/GenocchiNumber.html">Genocchi Number.</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Genocchi_number">Genocchi number</a>
%F 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.
%F 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.
%t 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]
%t (* or, from Seidel's triangle A014781: *)
%t 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}]
%Y Cf. A001469, A014781, A036968, A005439 (a(n) second and third diagonals), A164555/A027642, A209308, A226158, A240581(n)/A239315(n) (core of Bernoulli numbers).
%K sign
%O 0,2
%A _Jean-François Alcover_ and _Paul Curtz_, Nov 18 2016
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