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A172385 a(n) = 1 if n=0, otherwise Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k) *(-1)^k*a(n-1-2k).

%I A172385 #20 Feb 25 2023 09:39:23
%S A172385 1,1,1,0,-2,-4,-1,14,34,2,-189,-439,263,3796,6997,-14437,-96643,
%T A172385 -106774,671097,2800836,57519,-31088662,-82674287,155322877,
%U A172385 1455331563,1936970102,-14267868745,-66446614533,19215003803,1037638182571,2654391633166,-8675836955120,-67833031653088
%N A172385 a(n) = 1 if n=0, otherwise Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k) *(-1)^k*a(n-1-2k).
%H A172385 Seiichi Manyama, <a href="/A172385/b172385.txt">Table of n, a(n) for n = 0..949</a>
%F A172385 G.f.: A(x) = 1 + (x/(1+x^2))*A(x/(1+x^2)).
%e A172385 Eigensequence for the number triangle
%e A172385    1;
%e A172385    1,   0;
%e A172385    0,   1,   0;
%e A172385   -1,   0,   1,   0;
%e A172385    0,  -2,   0,   1,   0;
%e A172385    1,   0,  -3,   0,   1,   0;
%e A172385    0,   3,   0,  -4,   0,   1,   0;
%e A172385   -1,   0,   6,   0,  -5,   0,   1,   0;
%e A172385    0,  -4,   0,  10,   0,  -6,   0,   1,   0;
%e A172385    1,   0, -10,   0,  15,   0,  -7,   0,   1,   0;
%e A172385    0,   5,   0, -20,   0,  21,   0,  -8,   0,   1,   0;
%e A172385 (augmented version of Riordan array (1/(1+x^2), x/(1+x^2)).
%t A172385 a[n_] := If[n == 0, 1, Sum[Binomial[n - k - 1, k]*(-1)^k*a[n - 2*k - 1], {k, 0, Floor[(n - 1)/2]}]]; Table[a[n], {n, 0, 30}] (* _G. C. Greubel_, Oct 07 2018 *)
%Y A172385 Cf. A172383, A360898, A360899.
%K A172385 easy,sign
%O A172385 0,5
%A A172385 _Paul Barry_, Feb 01 2010
%E A172385 Name corrected by _G. C. Greubel_, Oct 07 2018
%E A172385 Terms a(27) and beyond from _Seiichi Manyama_, Feb 25 2023