A366356 - OEIS

%I #23 Oct 20 2023 06:45:06

%S 1,2,-1,6,-17,71,-292,1284,-5807,26961,-127627,613815,-2990680,

%T 14730714,-73229290,366936232,-1851352819,9397497759,-47957377933,

%U 245903408245,-1266266092111,6545667052321,-33954266444497,176689391245147,-922112642288148,4825154135801698

%N G.f. satisfies A(x) = 1/(1 - x) + x/A(x).

%H Paolo Xausa, <a href="/A366356/b366356.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: A(x) = -2*x*(1-x) / (1-sqrt(1+4*x*(1-x)^2)).

%F a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(2*k-1,k) * binomial(2*k-1,n-k)/(2*k-1).

%t A366356[n_]:=(-1)^(n-1)Sum[Binomial[2k-1,k]Binomial[2k-1,n-k]/(2k-1),{k,0,n}];

%t Array[A366356,30,0] (* _Paolo Xausa_, Oct 20 2023 *)

%o (PARI) a(n) = (-1)^(n-1)*sum(k=0, n, binomial(2*k-1, k)*binomial(2*k-1, n-k)/(2*k-1));

%Y Partial sums give A363816.

%Y Cf. A007317, A199475, A349289, A349290, A349291, A349292, A349293, A366357, A366358, A366359.

%Y Cf. A112478, A366363.

%K sign

%O 0,2

%A _Seiichi Manyama_, Oct 08 2023