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A188312 Expansion of (1/(1-x^2))*c(x/((1-x)*(1-x^2))) where c(x) is the g.f. of A000108.

%I A188312 #29 Oct 19 2024 21:40:19
%S A188312 1,1,4,12,45,174,709,2978,12825,56303,251060,1133943,5176926,23851690,
%T A188312 110759081,517853840,2435786531,11517940357,54722081630,261089977806,
%U A188312 1250479470053,6009884614944,28975052979797,140098515402139,679189779433800,3300702453217325,16076773046682690
%N A188312 Expansion of (1/(1-x^2))*c(x/((1-x)*(1-x^2))) where c(x) is the g.f. of A000108.
%C A188312 Hankel transform is the (25,-29) Somos-4 sequence A188313. Image of Catalan numbers by A188316.
%H A188312 G. C. Greubel, <a href="/A188312/b188312.txt">Table of n, a(n) for n = 0..1000</a>
%F A188312 G.f.: (1-x^2 - sqrt(1-4*x-6*x^2+x^4))/(2*x*(1+x)).
%F A188312 G.f.: u(x)=1/(1-x^2-x/(1-x-x*u(x))).
%F A188312 G.f.: 1/(1-x^2-x/(1-x-x/(1-x^2-x/(1-x-x/(1-...))))) (continued fraction).
%F A188312 Conjecture: (n+1)*a(n) +(3-4*n)*a(n-1) + (7-6*n)*a(n-2) -a(n-3) +(n-4)*a(n-4)=0. - _R. J. Mathar_, Nov 15 2011
%F A188312 a(n) = a(n-1) + (-1)^n + Sum_{i=0..n-1} a(i)*a(n-1-i). - _Vladimir Kruchinin_, May 03 2018
%F A188312 a(n) = Sum_{i=0..n} Sum_{k=1..n-i} (-1)^(n-k-i)*C(k+i-1,k-1)*C(2*k+i-2,k+i-1)*C(n-i-1,n-k-i)/k. - _Vladimir Kruchinin_, May 03 2018
%F A188312 a(n) = Sum_{i=0..n} (-1)^(n-i)*hypergeom([(i+1)/2, i/2+1, i-n], [1, 2], 4). - _Peter Luschny_, May 03 2018
%p A188312 a := n -> add((-1)^(n-i)*hypergeom([(i+1)/2, i/2+1, i-n], [1, 2], 4), i=0..n);
%p A188312 seq(simplify(a(n)), n=0..26); # _Peter Luschny_, May 03 2018
%t A188312 CoefficientList[Series[(1-x^2 -Sqrt[1-4*x-6*x^2+x^4])/(2*x*(1+x)), {x, 0, 50}], x] (* _G. C. Greubel_, Aug 14 2018 *)
%o A188312 (Maxima)
%o A188312 a(n):=sum(sum((-1)^(n-k-i)*binomial(k+i-1, k-1)*binomial(2*k+i-2, k+i-1)* binomial(n-i-1, n-k-i)/k,k,1,n-i),i,0,n); /* _Vladimir Kruchinin_, May 03 2018 */
%o A188312 (PARI) x='x+O('x^50); Vec((1-x^2 -sqrt(1-4*x-6*x^2+x^4))/(2*x*(1+x))) \\ _G. C. Greubel_, Aug 14 2018
%o A188312 (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-x^2 -Sqrt(1-4*x-6*x^2+x^4))/(2*x*(1+x)))); // _G. C. Greubel_, Aug 14 2018
%Y A188312 Cf. A188314.
%K A188312 nonn
%O A188312 0,3
%A A188312 _Paul Barry_, Mar 28 2011