A002317 Related to Genocchi numbers.

2, 5, 7, -26, -265, -1351, -5042, -13775, -18817, 70226, 716035, 3650401, 13623482, 37220045, 50843527, -189750626, -1934726305, -9863382151, -36810643322, -100568547815, -137379191137, 512706121226, 5227629760075, 26650854921601

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy

Comments

Denoted by beta'_n by Lehmer.

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.
• Index entries for two-way infinite sequences
• Index entries for linear recurrences with constant coefficients, signature (6,-11,-6,-1).

Crossrefs

a(n) = (-1)^n*A002316(-1-n).

Programs

Mathematica

a[0] = 2; a[1] = 5; a[2] = 7; a[3] = -26; a[n_] := a[n] = -a[n-4] - 6*a[n-3] - 11*a[n-2] + 6*a[n-1]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, May 23 2013 *)
CoefficientList[Series[(2 - 7 x - x^2 - x^3) / (1 - 6 x + 11 x^2 + 6 x^3 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 21 2013 *)
LinearRecurrence[{6,-11,-6,-1},{2,5,7,-26},40] (* Harvey P. Dale, Jun 04 2017 *)

PARI

{a(n)=if(n>=0, polcoeff( (2-7*x-x^2-x^3)/(1-6*x+11*x^2+6*x^3+x^4) +x*O(x^n),n), n=-1-n; (-1)^n*polcoeff( (1-x+7*x^2+2*x^3)/(1-6*x+11*x^2+6*x^3+x^4) +x*O(x^n),n) )} /* Michael Somos, Mar 27 2005 */

Formula

G.f.: (2-7*x-x^2-x^3)/(1-6*x+11*x^2+6*x^3+x^4).

a(n) = -2702*a(n-6) - a(n-12).