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A188482 Diagonal sums of the Riordan matrix (1/(1-4x),(1-sqrt(1-4x))/(2*sqrt(1-4x))) (A188481).

%I A188482 #14 Dec 26 2023 10:07:18
%S A188482 1,4,17,71,295,1221,5040,20761,85380,350659,1438568,5896098,24145941,
%T A188482 98812861,404118745,1651811920,6748282361,27556753703,112482005583,
%U A188482 458958881572,1872034052651,7633342954234,31116252892098,126806214027741,516633711969649
%N A188482 Diagonal sums of the Riordan matrix (1/(1-4x),(1-sqrt(1-4x))/(2*sqrt(1-4x))) (A188481).
%F A188482 a(n) = [x^n] 1/((1-x)^(n+1)*(1-2*x-x^2+x^3)).
%F A188482 a(n) = Sum_{k=0..n} Sum_{i=0..k} binomial(k,i)*binomial(2*n+k-2*i+2, n-k)*(-1)^i.
%F A188482 G.f.: (2 - 7*x - 4*x^2 + x*sqrt(1-4*x))/(2 - 14*x + 16*x^2 + 30*x^3 + 8*x^4).
%F A188482 Conjecture: (-n+1)*a(n) + (7*n-9)*a(n-1) + 2*(-4*n+7)*a(n-2) + (-15*n+23)*a(n-3) + 2*(-2*n+3)*a(n-4) = 0. - _R. J. Mathar_, Jun 14 2016
%t A188482 Table[Sum[Binomial[k,i]Binomial[2n+k-2i+2,n-k](-1)^i,{k,0,n},{i,0,k}],{n,0,12}]
%o A188482 (Maxima) makelist(sum(sum(binomial(k,i)*binomial(2*n+k-2*i+2,n-k)*(-1)^i,i,0,k),k,0,n),n,0,12);
%Y A188482 Cf. A141223, A188481.
%K A188482 nonn,easy
%O A188482 0,2
%A A188482 _Emanuele Munarini_, Apr 01 2011