A278331 Shifted sequence of second differences of Genocchi numbers.

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

Author

Jean-François Alcover and Paul Curtz, Nov 18 2016

Keywords

• sign

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

Crossrefs

Cf. A001469, A014781, A036968, A005439 (a(n) second and third diagonals), A164555/A027642, A209308, A226158, A240581(n)/A239315(n) (core of Bernoulli numbers).

Programs

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}]

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.