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A002316 Related to Bernoulli numbers.
(Formerly M3941 N1624)

%I M3941 N1624

%S 1,5,26,97,265,362,-1351,-13775,-70226,-262087,-716035,-978122,

%T 3650401,37220045,189750626,708158977,1934726305,2642885282,

%U -9863382151,-100568547815,-512706121226,-1913445293767,-5227629760075,-7141075053842

%N Related to Bernoulli numbers.

%C Denoted by beta_n by Lehmer.

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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H Robert Israel, <a href="/A002316/b002316.txt">Table of n, a(n) for n = 0..1746</a>

%H D. H. Lehmer, <a href="http://www.jstor.org/stable/1968647">Lacunary recurrence formulas for the numbers of Bernoulli and Euler</a>, Annals Math., 36 (1935), 637-649.

%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>

%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,-6,-1).

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

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

%F 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

%p 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):

%p map(f, [$0..25]); # _Robert Israel_, Aug 23 2017

%t 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 *)

%o (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 */

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

%K sign,easy

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, Dec 23 1999

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Last modified May 18 03:03 EDT 2021. Contains 343994 sequences. (Running on oeis4.)