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A278331
Shifted sequence of second differences of Genocchi numbers.
1
0, -2, -2, 6, 14, -34, -138, 310, 1918, -4146, -36154, 76454, 891342, -1859138, -27891050, 57641238, 1080832286, -2219305810, -50833628826, 103886563462, 2853207760750, -5810302084962, -188424521441482, 382659344967926, 14464296482284734, -29311252309537394, -1277229462293249018
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
Cf. A001469, A014781, A036968, A005439 (a(n) second and third diagonals), A164555/A027642, A209308, A226158, A240581(n)/A239315(n) (core of Bernoulli numbers).
Sequence in context: A248096 A002203 A300863 * A097341 A142710 A369608
KEYWORD
sign
AUTHOR
STATUS
approved