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A085455 Sum_{i=0..n} Sum_{j=0..i} a(j) * a(i-j) = (-3)^n.

%I A085455 #19 Aug 22 2025 10:55:36
%S A085455 1,-2,4,-10,26,-70,192,-534,1500,-4246,12092,-34606,99442,-286730,
%T A085455 829168,-2403834,6984234,-20331558,59287740,-173149662,506376222,
%U A085455 -1482730098,4346486256,-12754363650,37461564504,-110125172682,323990062452,-953883382354,2810310510110,-8284915984726
%N A085455 Sum_{i=0..n} Sum_{j=0..i} a(j) * a(i-j) = (-3)^n.
%H A085455 Seiichi Manyama, <a href="/A085455/b085455.txt">Table of n, a(n) for n = 0..1000</a>
%F A085455 G.f.: A(x)=Sqrt((1-x)/(1+3x)).
%F A085455 G.f.: G(0), where G(k)= 1 + 4*x*(4*k+1)/( (x-1)*(4*k+2) - x*(x-1)*(4*k+2)*(4*k+3)/(x*(4*k+3) + (x-1)*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 26 2013
%F A085455 From _Seiichi Manyama_, Feb 03 2023: (Start)
%F A085455 a(n) = Sum_{k=0..n} (-1)^k * binomial(n-1,n-k) * binomial(2*k,k).
%F A085455 n*a(n) = -2*n*a(n-1) + 3*(n-2)*a(n-2). (End)
%F A085455 From _Seiichi Manyama_, Aug 22 2025: (Start)
%F A085455 a(n) = (1/4)^n * Sum_{k=0..n} (-3)^k * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
%F A085455 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)
%t A085455 CoefficientList[Series[Sqrt[(1-x)/(1+3x)], {x, 0, 30}], x]
%o A085455 (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n-1, n-k)*binomial(2*k, k)); \\ _Seiichi Manyama_, Feb 03 2023
%Y A085455 Absolute values are in A025565.
%Y A085455 Cf. A085362, A085363, A085364.
%K A085455 easy,sign,changed
%O A085455 0,2
%A A085455 Mario Catalani (mario.catalani(AT)unito.it), Jul 01 2003