cp's OEIS Frontend

A143338 G.f. A(x) satisfies A(x) = 1 + x*A(x)^3*A(-x).

%I A143338 #12 Jul 08 2025 07:42:07
%S A143338 1,1,2,8,26,127,478,2536,10250,56900,239880,1370272,5940054,34607146,
%T A143338 153018932,904441648,4058644842,24254529036,110096276440,663665021280,
%U A143338 3040205250984,18455364854839,85176971647470,520059936017128
%N A143338 G.f. A(x) satisfies A(x) = 1 + x*A(x)^3*A(-x).
%F A143338 a(0) = 1; a(n) = Sum_{i, j, k, l>=0 and i+j+k+l=n-1} (-1)^i * a(i) * a(j) * a(k) * a(l). - _Seiichi Manyama_, Jul 08 2025
%e A143338 G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 26*x^4 + 127*x^5 + 478*x^6 +...
%e A143338 Compare bisections of A(x)^2, A(x)^2*A(-x), and A(x)^4*A(-x)^2:
%e A143338 A(x)^2 = 1 + 2*x + 5*x^2 + 20*x^3 + 72*x^4 + 338*x^5 + 1378*x^6 + 6952*x^7 +...
%e A143338 A(x)^2*A(-x) = 1 + x + 5*x^2 + 11*x^3 + 72*x^4 + 191*x^5 + 1378*x^6 + 3979*x^7 +...
%e A143338 A(x)^4*A(-x)^2 = 1 + 2*x + 11*x^2 + 32*x^3 + 191*x^4 + 636*x^5 + 3979*x^6 +...
%e A143338 Related expansions:
%e A143338 A(x)^3 = 1 + 3*x + 9*x^2 + 37*x^3 + 144*x^4 + 669*x^5 + 2882*x^6 + 14229*x^7 +...
%e A143338 A(x)^3*A(-x) = 1 + 2*x + 8*x^2 + 26*x^3 + 127*x^4 + 478*x^5 + 2536*x^6 +...
%e A143338 A(x)^3*A(-x)^2 = 1 + x + 8*x^2 + 14*x^3 + 127*x^4 + 264*x^5 + 2536*x^6 +...
%o A143338 (PARI) {a(n)=local(A=1+x+O(x^21));for(i=0,n,A=1+x*A^3*subst(A,x,-x));polcoeff(A,n)}
%Y A143338 Cf. A047749, A143549, A143551.
%Y A143338 Cf. A143546.
%K A143338 nonn
%O A143338 0,3
%A A143338 _Paul D. Hanna_, Aug 09 2008