A393196 - OEIS

%I #17 Feb 05 2026 21:02:27

%S 1,-1,-1,3,-1,0,3,-14,-1,21,0,6,3,-56,-14,102,-1,-63,21,-225,0,552,6,

%T -198,3,-791,-56,1428,-14,-1200,102,-1398,-1,6009,-63,-3819,21,-7344,

%U -225,12768,0,-8184,552,-8164,6,45288,-198,-37794,3,-64547,-791,108915,-56,-33768,1428,-59916,-14

%N G.f. A(x) satisfies A(x) = A(x^2) - x*A(x^2)^3.

%H Seiichi Manyama, <a href="/A393196/b393196.txt">Table of n, a(n) for n = 0..1000</a>

%F a(0) = 1, a(2*n) = a(n), a(2*n+1) = -Sum_{i,j,k>=0 and i+j+k=n} a(i) * a(j) * a(k).

%t n=60; A=Series[1,{x,0,n}]; Do[A=Normal@Series[(A/. x->x^2)-x*(A/. x->x^2)^3,{x,0,n}],{6}];

%t CoefficientList[A,x] (* _Vincenzo Librandi_, Feb 05 2026 *)

%o (Ruby)

%o def A393196(n)

%o   ary = [1]

%o   (1..n).each{|i|

%o     m = i / 2

%o     if i.even?

%o       ary << ary[m]

%o     else

%o       s = 0

%o       (0..m).each{|j|

%o         (0..m - j).each{|k|

%o           s += ary[j] * ary[k] * ary[m - j - k]

%o         }

%o       }

%o       ary << -s

%o     end

%o   }

%o   ary

%o end

%o p A393196(60)

%o (Magma) N := 60; R<x> := PowerSeriesRing(Integers()); A := R!1; for i in [1..6] do

%o  B := ChangePrecision(Evaluate(A, x^2), N+1); A := ChangePrecision(B - x*B^3, N+1);

%o end for; [ Coefficient(A, i) : i in [0..N] ]; // _Vincenzo Librandi_, Feb 05 2026

%Y Cf. A374571.

%K sign

%O 0,4

%A _Seiichi Manyama_, Feb 05 2026