A393196 G.f. A(x) satisfies A(x) = A(x^2) - x*A(x^2)^3.

1, -1, -1, 3, -1, 0, 3, -14, -1, 21, 0, 6, 3, -56, -14, 102, -1, -63, 21, -225, 0, 552, 6, -198, 3, -791, -56, 1428, -14, -1200, 102, -1398, -1, 6009, -63, -3819, 21, -7344, -225, 12768, 0, -8184, 552, -8164, 6, 45288, -198, -37794, 3, -64547, -791, 108915, -56, -33768, 1428, -59916, -14

Offset

0,4

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

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).

Mathematica

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}];
CoefficientList[A, x] (* Vincenzo Librandi, Feb 05 2026 *)

Prog

(Ruby)
def A393196(n)
  ary = [1]
  (1..n).each{|i|
    m = i / 2
    if i.even?
      ary << ary[m]
    else
      s = 0
      (0..m).each{|j|
        (0..m - j).each{|k|
          s += ary[j] * ary[k] * ary[m - j - k]
        }
      }
      ary << -s
    end
  }
  ary
end
p A393196(60)
(Magma) N := 60; R<x> := PowerSeriesRing(Integers()); A := R!1; for i in [1..6] do
 B := ChangePrecision(Evaluate(A, x^2), N+1); A := ChangePrecision(B - x*B^3, N+1);
end for; [ Coefficient(A, i) : i in [0..N] ]; // Vincenzo Librandi, Feb 05 2026

Crossrefs

Cf. A374571.

Sequence in context: A384486 A166407 A285123 * A159059 A340262 A346369

Adjacent sequences: A393193 A393194 A393195 * A393197 A393198 A393199

Keyword

sign

Author

Seiichi Manyama, Feb 05 2026

Status

approved