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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 (list; table; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..26.

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 A014431

Adjacent sequences:  A278328 A278329 A278330 * A278332 A278333 A278334

KEYWORD

sign,tabl

AUTHOR

Jean-Fran├žois Alcover and Paul Curtz, Nov 18 2016

STATUS

approved

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Last modified November 14 07:07 EST 2019. Contains 329110 sequences. (Running on oeis4.)