A002316 Related to Bernoulli numbers.

1, 5, 26, 97, 265, 362, -1351, -13775, -70226, -262087, -716035, -978122, 3650401, 37220045, 189750626, 708158977, 1934726305, 2642885282, -9863382151, -100568547815, -512706121226, -1913445293767, -5227629760075, -7141075053842

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy

Comments

Denoted by beta_n by Lehmer.

References

• B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 84.
• 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

• Robert Israel, Table of n, a(n) for n = 0..1746
• 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 sequences related to Bernoulli numbers.
• Index entries for linear recurrences with constant coefficients, signature (6,-11,-6,-1).

Crossrefs

a(n) = (-1)^n*A002317(-1-n).

Programs

Maple

f:= gfun:-rectoproc({a(0)=1, a(1)=5, a(2)=26, a(3)=97, a(n)=6*a(n-1)-11*a(n-2)-6*a(n-3)-a(n-4)},a(n),remember):
map(f, [$0..25]); # Robert Israel, Aug 23 2017

Mathematica

LinearRecurrence[{6,-11,-6,-1},{1,5,26,97},30] (* or *) CoefficientList[ Series[(2x^3+7x^2-x+1)/(x^4+6x^3+11x^2-6x+1),{x,0,30}],x] (* Harvey P. Dale, Jun 13 2011 *)

PARI

{a(n)=if(n>=0, 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), n=-1-n; (-1)^n*polcoeff( (2-7*x-x^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

a(0)..a(11) are as given (with signs); for n >= 12, a(n) = -2702*a(n-6) - a(n-12).

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

a(0)=1, a(1)=5, a(2)=26, a(3)=97, a(n) = 6*a(n-1) - 11*a(n-2) - 6*a(n-3) - a(n-4). - Harvey P. Dale, Jun 13 2011

Extensions

More terms from James Sellers, Dec 23 1999