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A277969 a(n) = Sum_{k=0..n} binomial(n-3,n-k)*Catalan(k).

%I A277969 #24 Oct 23 2018 19:34:25
%S A277969 1,-1,2,5,19,75,305,1270,5390,23236,101480,448085,1997115,8973255,
%T A277969 40602093,184853055,846206025,3892585325,17984308775,83417287855,
%U A277969 388297304825,1813341109825,8493372326675,39889629750600,187812852106636
%N A277969 a(n) = Sum_{k=0..n} binomial(n-3,n-k)*Catalan(k).
%H A277969 Robert Israel, <a href="/A277969/b277969.txt">Table of n, a(n) for n = 0..1430</a>
%F A277969 G.f.: ((1-x)^3*(1-sqrt((5*x-1)/(x-1))))/(2*x).
%F A277969 a(n) ~ 8*5^(n-3/2) / (sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Nov 07 2016
%F A277969 (5*n-10)*a(n)-(7+6*n)*a(n+1)+(n+3)*a(n+2)=0 for n >= 2. - _Robert Israel_, Nov 21 2016
%F A277969 a(n) = A055452(n+1) for n > 2. - _Georg Fischer_, Oct 23 2018
%p A277969 f:= gfun:-rectoproc({(5*n-10)*a(n)+(-7-6*n)*a(n+1)+(n+3)*a(n+2),a(0) = 1, a(1) = -1, a(2) = 2, a(3) = 5},a(n),remember):
%p A277969 map(f, [$0..30]); # _Robert Israel_, Nov 21 2016
%t A277969 CoefficientList[Series[((1 - x)^3 (1 - Sqrt[(5 x - 1) / (x - 1)])) / (2 x), {x, 0, 25}], x] (* _Vincenzo Librandi_, Nov 07 2016 *)
%o A277969 (Maxima)
%o A277969 a(n):=sum((binomial(2*k,k)*binomial(n-3,n-k))/(k+1),k,0,n);
%o A277969 (PARI) x='x+O('x^50); Vec(((1-x)^3*(1-sqrt((5*x-1)/(x-1))))/(2*x)) \\ _G. C. Greubel_, Apr 09 2017
%Y A277969 Cf. A000108, A055452.
%K A277969 sign
%O A277969 0,3
%A A277969 _Vladimir Kruchinin_, Nov 06 2016