A196022 E.g.f. satisfies: A(x) = Sum_{n>=0} 1/n! * Sum_{k=0..n} C(n,k)*(1 + x/2*A(x)^(n-k))^(n-k) * (-1 + x/2*A(x)^k)^k.
1, 1, 3, 22, 269, 4676, 106027, 2966909, 98814921, 3818572219, 167999566061, 8292669672343, 453958089569833, 27300462412168531, 1789601591598075179, 127031469572649790786, 9709396866317453042033, 795229841271312470470904, 69499103185446754949076817
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 269*x^4/4! + 4676*x^5/5! +... where e.g.f. A = A(x) satisfies: A(x) = 1 + ((1+x/2*A) + (-1+x/2*A)) + 1/2!*((1+x/2*A^2)^2 + 2*(1+x/2*A)*(-1+x/2*A) + (-1+x/2*A^2)^2) + 1/3!*((1+x/2*A^3)^3 + 3*(1+x/2*A^2)^2*(-1+x/2*A) + 3*(1+x/2*A)*(-1+x/2*A^2)^2 + (-1+x/2*A^3)^3) + 1/4!*((1+x/2*A^4)^4 + 4*(1+x/2*A^3)^3*(-1+x/2*A) + 6*(1+x/2*A^2)^2*(-1+x/2*A^2)^2 + 4*(1+x/2*A)*(-1+x/2*A^3)^3 + (-1+x/2*A^4)^4) + 1/5!*((1+x/2*A^5)^5 + 5*(1+x/2*A^4)^4*(-1+x/2*A) + 10*(1+x/2*A^3)^3*(-1+x/2*A^2)^2 + 10*(1+x/2*A^2)^2*(-1+x/2*A^3)^3 + 5*(1+x/2*A)*(-1+x/2*A^4)^4 + (-1+x/2*A^5)^5) +...
Programs
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PARI
{a(n)=local(A=1+x,X=x+x*O(x^n));for(i=1,n,A=1+sum(m=1,n,1/m!*sum(k=0,m,binomial(m,k)*(1+X/2*A^(m-k))^(m-k)*(-1+X/2*A^k)^k)));n!*polcoeff(A,n)}