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A099621 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k+1) * 3^(n-k-1)*(4/3)^k.

%I A099621 #12 Sep 08 2022 08:45:15
%S A099621 0,1,6,31,144,637,2730,11467,47508,194953,794574,3222583,13023192,
%T A099621 52491349,211161138,848231779,3403688796,13647040225,54685016022,
%U A099621 219030629455,876994213920,3510591943981,14050213040826,56224387958011
%N A099621 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k+1) * 3^(n-k-1)*(4/3)^k.
%C A099621 In general a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k+1) * u^(n-k-1)* (v/u)^(k-1) has g.f. x^2/((1-u*x)*(1-u*x-v*x^2)) and satisfies the recurrence a(n) = 2*u*a(n-1) - (u^2-v)*a(n-2) - u*v*a(n-3).
%H A099621 G. C. Greubel, <a href="/A099621/b099621.txt">Table of n, a(n) for n = 0..1000</a>
%H A099621 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-5,-12).
%F A099621 G.f.: x^2/((1-3*x)*(1-3*x-4*x^2)).
%F A099621 a(n) = 6*a(n-1) - 5*a(n-2) - 12*a(n-3).
%F A099621 From _G. C. Greubel_, Jun 06 2019: (Start)
%F A099621 a(n) = (4^(n+2) - 5*3^(n+1) - (-1)^n)/20.
%F A099621 E.g.f.: (-exp(-x) - 15*exp(3*x) + 16*exp(4*x))/20. (End)
%t A099621 Table[Sum[Binomial[n-k,k+1]3^(n-k-1) (4/3)^k,{k,0,Floor[n/2]}],{n,0,25}] (* or *) LinearRecurrence[{6,-5,-12},{0,1,6},30] (* _Harvey P. Dale_, Dec 13 2012 *)
%t A099621 Table[(4^(n+2)-5*3^(n+1)-(-1)^n)/20, {n,0,30}] (* _G. C. Greubel_, Jun 06 2019 *)
%o A099621 (PARI) vector(30, n, n--; (4^(n+2)-5*3^(n+1)-(-1)^n)/20) \\ _G. C. Greubel_, Jun 06 2019
%o A099621 (Magma) [(4^(n+2)-5*3^(n+1)-(-1)^n)/20: n in [0..30]]; // _G. C. Greubel_, Jun 06 2019
%o A099621 (Sage) [(4^(n+2)-5*3^(n+1)-(-1)^n)/20 for n in (0..30)] # _G. C. Greubel_, Jun 06 2019
%o A099621 (GAP) List([0..30], n-> (4^(n+2)-5*3^(n+1)-(-1)^n)/20) # _G. C. Greubel_, Jun 06 2019
%Y A099621 Cf. A094705, A099622.
%K A099621 easy,nonn
%O A099621 0,3
%A A099621 _Paul Barry_, Oct 25 2004