A222589 G.f. satisfies: A(x) = Sum_{n>=0} x^n*(1 + n*x)^n * A(x)^n / (1 + x*A(x) + n*x^2*A(x))^n.

1, 1, 2, 5, 14, 41, 128, 409, 1355, 4564, 15728, 54904, 194740, 698042, 2532483, 9270351, 34268276, 127677731, 479723132, 1815553953, 6923744832, 26587139445, 102838915279, 400513959602, 1571152132075, 6206954038519, 24705172805012, 99071049959707, 400475021255313

Offset: 0

Author

Paul D. Hanna, Feb 25 2013

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 41*x^5 + 128*x^6 + 409*x^7 +...
where:
A(x) = 1 + x*(1+x)*A(x)/(1+x*(1+x)*A(x)) + x^2*(1+2*x)^2*A(x)^2/(1+x*(1+2*x)*A(x))^2 + x^3*(1+3*x)^3*A(x)^3/(1+x*(1+3*x)*A(x))^3 + x^4*(1+4*x)^4*A(x)^4/(1+x*(1+4*x)*A(x))^4 +...
Also,
A(x) = 1/2 + (1 + 2*x*A(x))/2 * (1 + 2*x^2*A(x) + 6*x^4*A(x)^2 + 24*x^6*A(x)^3 + 120*x^8*A(x)^4 + 720*x^10*A(x)^5 + 5040*x^12*A(x)^6 +...).
		

Crossrefs

Cf. A187741.

Programs

PARI

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0, n, (x+m*x^2)^m*A^m / (1 + x*A+m*x^2*A +x*O(x^n))^m));polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

PARI

{a(n)=local(A=1+x);for(i=1,n,A=1/2+(1+2*x*A)*sum(k=0,n,(k+1)!/2*x^(2*k)*(A+x*O(x^n))^k));polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. satisfies: A(x) = 1/2 + (1 + 2*x*A(x))/2 * Sum_{n>=0} (n+1)! * x^(2*n) * A(x)^n.