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A247526 a(n) = L(n+1) * L(n) * L(n-1) * L(n-2) / 6, where L(n) = Lucas numbers (A000032).

%I A247526 #31 Sep 08 2022 08:46:09
%S A247526 -1,-1,4,14,154,924,6699,44979,310764,2123554,14571974,99833524,
%T A247526 684385079,4690541639,32150245204,220358978774,1510368355474,
%U A247526 10352204457804,70955102255139,486333408161979,3333379024971324,22847319059525674,156597856242950654
%N A247526 a(n) = L(n+1) * L(n) * L(n-1) * L(n-2) / 6, where L(n) = Lucas numbers (A000032).
%H A247526 G. C. Greubel, <a href="/A247526/b247526.txt">Table of n, a(n) for n = 0..1000</a>
%H A247526 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,15,-15,-5,1).
%F A247526 G.f.: -(1 - 4*x - 24*x^2 + 6*x^3 - 4*x^4)/((1-x)*(1 + 3*x + x^2)*(1 - 7*x + x^2)).
%F A247526 a(n) = a(1-n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5) for all n in Z.
%F A247526 0 = a(n)*(+a(n+1) - 2*a(n+2)) + a(n+1)*(-5*a(n+1) + a(n+2)) for all n in Z.
%F A247526 From _Klaus Purath_, Oct 02 2020: (Start)
%F A247526 a(n) = (L(n-2)*L(n-1)^3 - L(n-1)*L(n-2)^3)/6 where L = Lucas.
%F A247526 a(n) = f(n-3)*f(n-2)*f(n-1)*f(n)/96 where f = A022112.
%F A247526 a(n) = (f(n-2)*f(n-1)^3 - f(n-1)*f(n-2)^3)/96 where f = A022112.
%F A247526 (End)
%F A247526 2*a(n) = A098149(n) +A004187(n+1)-6*A004187(n) -2 . - _R. J. Mathar_, Sep 24 2021
%e A247526 G.f. = -1 - x + 4*x^2 + 14*x^3 + 154*x^4 + 924*x^5 + 6699*x^6 + 44979*x^7 + ...
%t A247526 CoefficientList[Series[-(1-4*x-24*x^2+6*x^3-4*x^4)/((1-x)*(1+3*x+x^2)*(1 - 7*x+x^2)), {x, 0, 60}], x] (* _G. C. Greubel_, Aug 05 2018 *)
%t A247526 Times@@#/6&/@Partition[LucasL[Range[-2,30]],4,1] (* or *) LinearRecurrence[{5,15,-15,-5,1},{-1,-1,4,14,154},30] (* _Harvey P. Dale_, Apr 20 2022 *)
%o A247526 (PARI) {a(n) = my(u = fibonacci(n), v = fibonacci(n-1)); (3*u + v) * (u + 2*v) * (2*u - v) * (-u + 3*v) / 6};
%o A247526 (PARI) {a(n) = if( n<1, n=1-n); polcoeff( - (1 - 4*x - 24*x^2 + 6*x^3 - 4*x^4) / ((1 - x) * (1 + 3*x + x^2) * (1 - 7*x +x^2)) + x * O(x^n), n)};
%o A247526 (Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(-(1-4*x-24*x^2+6*x^3-4*x^4)/((1-x)*(1+3*x+x^2)*(1 - 7*x+x^2)))); // _G. C. Greubel_, Aug 05 2018
%Y A247526 Cf. A000032, A085696.
%K A247526 sign,easy
%O A247526 0,3
%A A247526 _Michael Somos_, Sep 19 2014