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%I #25 Mar 30 2023 20:54:49
%S 1,7,50,363,2669,19814,148153,1113615,8402722,63577171,481991621,
%T 3659227062,27808295345,211479529943,1609093780114,12247558413819,
%U 93245414394973,710040492168070,5407464407991017,41185377124992351,313703861897268866,2389549742539808867
%N Sixth binomial transform of F(n+1).
%C Binomial transform of A081570.
%C Case k=6 of family of recurrences a(n) = (2k+1)*a(n-1) - A028387(k-1)*a(n-2), a(0)=1, a(1)=k+1.
%H Vincenzo Librandi, <a href="/A081571/b081571.txt">Table of n, a(n) for n = 0..200</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (13,-41).
%F a(n) = 13*a(n-1) - 41*a(n-2), a(0)=1, a(1)=7.
%F a(n) = (1/2 - sqrt(5)/10)*(13/2 - sqrt(5)/2)^n + (sqrt(5)/10 + 1/2)*(sqrt(5)/2 + 13/2)^n.
%F G.f.: (1 - 6*x)/(1 - 13*x + 41*x^2).
%F a(n) = Sum_{k=0..n} A094441(n,k)*6^k. - _Philippe Deléham_, Dec 14 2009
%F E.g.f.: exp(13*x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5. - _Stefano Spezia_, Mar 30 2023
%p seq(coeff(series((1-6*x)/(1-13*x+41*x^2), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Aug 12 2019
%t CoefficientList[Series[(1-6x)/(1 -13x +41x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 09 2013 *)
%o (Magma) I:=[1, 7]; [n le 2 select I[n] else 13*Self(n-1)-41*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Aug 09 2013
%o (PARI) my(x='x+O('x^30)); Vec((1-6*x)/(1-13*x+41*x^2)) \\ _G. C. Greubel_, Aug 12 2019
%o (Sage)
%o def A081571_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P((1-6*x)/(1-13*x+41*x^2)).list()
%o A081571_list(30) # _G. C. Greubel_, Aug 12 2019
%o (GAP) a:=[1,7];; for n in [3..30] do a[n]:=13*a[n-1]-41*a[n-2]; od; a; # _G. C. Greubel_, Aug 12 2019
%Y Cf. A000045, A028387, A081570.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Mar 22 2003
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