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

1, 1, 2, 8, 26, 127, 478, 2536, 10250, 56900, 239880, 1370272, 5940054, 34607146, 153018932, 904441648, 4058644842, 24254529036, 110096276440, 663665021280, 3040205250984, 18455364854839, 85176971647470, 520059936017128

Offset: 0

Author

Paul D. Hanna, Aug 09 2008

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 26*x^4 + 127*x^5 + 478*x^6 +...
Compare bisections of A(x)^2, A(x)^2*A(-x), and A(x)^4*A(-x)^2:
A(x)^2 = 1 + 2*x + 5*x^2 + 20*x^3 + 72*x^4 + 338*x^5 + 1378*x^6 + 6952*x^7 +...
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 +...
A(x)^4*A(-x)^2 = 1 + 2*x + 11*x^2 + 32*x^3 + 191*x^4 + 636*x^5 + 3979*x^6 +...
Related expansions:
A(x)^3 = 1 + 3*x + 9*x^2 + 37*x^3 + 144*x^4 + 669*x^5 + 2882*x^6 + 14229*x^7 +...
A(x)^3*A(-x) = 1 + 2*x + 8*x^2 + 26*x^3 + 127*x^4 + 478*x^5 + 2536*x^6 +...
A(x)^3*A(-x)^2 = 1 + x + 8*x^2 + 14*x^3 + 127*x^4 + 264*x^5 + 2536*x^6 +...
		

Crossrefs

Cf. A047749, A143549, A143551.

Cf. A143546.

Programs

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)}

Formula

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