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%I M1341 N0514 #37 Oct 15 2023 00:00:03
%S 2,5,7,-26,-265,-1351,-5042,-13775,-18817,70226,716035,3650401,
%T 13623482,37220045,50843527,-189750626,-1934726305,-9863382151,
%U -36810643322,-100568547815,-137379191137,512706121226,5227629760075,26650854921601
%N Related to Genocchi numbers.
%C Denoted by beta'_n by Lehmer.
%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 Vincenzo Librandi, <a href="/A002317/b002317.txt">Table of n, a(n) for n = 0..1000</a>
%H D. H. Lehmer, <a href="https://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/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,-6,-1).
%F G.f.: (2-7*x-x^2-x^3)/(1-6*x+11*x^2+6*x^3+x^4).
%F a(n) = -2702*a(n-6) - a(n-12).
%t 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 *)
%t 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 *)
%t LinearRecurrence[{6,-11,-6,-1},{2,5,7,-26},40] (* _Harvey P. Dale_, Jun 04 2017 *)
%o (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 */
%Y a(n) = (-1)^n*A002316(-1-n).
%K sign,easy
%O 0,1
%A _N. J. A. Sloane_
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