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A271785 a(n) = Sum_{k=0..(n-1)/2} (n+2-k)*binomial(n-1-k,k).

%I A271785 #26 May 05 2020 17:55:57
%S A271785 0,3,4,9,16,30,54,97,172,303,530,922,1596,2751,4724,8085,13796,23478,
%T A271785 39858,67517,114140,192603,324454,545714,916536,1537275,2575204,
%U A271785 4308897,7201912,12025038,20058990,33430297,55667596,92622471,153992954,255842890
%N A271785 a(n) = Sum_{k=0..(n-1)/2} (n+2-k)*binomial(n-1-k,k).
%C A271785 This is a sibling to the expansions A001629(n+1) = Sum_{k=0..(n-1)/2} (n-k) *binomial(n-1-k,k) and A226432(n+3) = Sum_{k=0..(n-1)/2} (n+1-k) *binomial(n-1-k,k).
%H A271785 Colin Barker, <a href="/A271785/b271785.txt">Table of n, a(n) for n = 0..1000</a>
%H A271785 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).
%F A271785 G.f.: x*(3-2*x-2*x^2) / (1-x-x^2)^2.
%F A271785 a(n) = 3*A001629(n+1) -2*A001629(n) -2*A001629(n-1).
%F A271785 From _Colin Barker_, Apr 14 2016: (Start)
%F A271785 a(n) = (2^(-1-n)*(-24*sqrt(5)*((1-sqrt(5))^n-(1+sqrt(5))^n)+5*((1-sqrt(5))^(1+n)+(1+sqrt(5))^(1+n))*n))/25.
%F A271785 a(n) = 2*a(n-1)+a(n-2)-2*a(n-3)-a(n-4) for n>3. (End)
%F A271785 E.g.f.: (1/25)*(sqrt(5)*(5*x + 24)*sinh((sqrt(5)*x)/2) + 15*x*cosh((sqrt(5)*x)/2))*exp(x/2). - _Ilya Gutkovskiy_, Apr 14 2016
%F A271785 a(n) = A006355(n+1)+A001629(n+1). - _R. J. Mathar_, May 20 2016
%p A271785 A271785 := proc(n)
%p A271785     add( (n+2-k)*binomial(n-1-k,k),k=0..(n-1)/2) ;
%p A271785 end proc:
%t A271785 LinearRecurrence[{2,1,-2,-1},{0,3,4,9},40] (* _Harvey P. Dale_, May 05 2020 *)
%o A271785 (PARI) concat(0, Vec(x*(3-2*x-2*x^2)/(1-x-x^2)^2 + O(x^50))) \\ _Colin Barker_, Apr 14 2016
%Y A271785 Cf. A001629.
%K A271785 nonn,easy
%O A271785 0,2
%A A271785 _R. J. Mathar_, Apr 14 2016