A245389 - OEIS

%I #9 Jul 21 2018 19:32:11

%S 1,2,6,23,102,496,2570,13959,78682,457243,2727360,16647048,103759186,

%T 659500772,4271197824,28175622291,189321228022,1296246842443,

%U 9049626101836,64481397834665,469461395956168,3497006117588399,26688813841105524,208977790442594368,1680981707733908594

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

%H Paul D. Hanna, <a href="/A245389/b245389.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. A(x) satisfies:

%F (1) A(x) = Sum_{n>=0} x^n / (1 - (n+1)*x*A(x)).

%F (2) A(x) = Sum_{n>=0} n! * x^n/(1-x)^(n+1) * A(x)^n / Product_{k=1..n} (1 + k*x*A(x)).

%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 23*x^3 + 102*x^4 + 496*x^5 + 2570*x^6 +...

%e where we have the following series identity:

%e A(x) = 1/(1-x*A(x)) + x/(1-2*x*A(x)) + x^2/(1-3*x*A(x)) + x^3/(1-4*x*A(x)) + x^4/(1-5*x*A(x)) + x^5/(1-6*x*A(x)) + x^6/(1-7*x*A(x)) +...

%e is equal to

%e A(x) = 1/(1-x) + x/(1-x)^2*A(x)/(1+x*A(x)) + 2!*x^2/(1-x)^3*A(x)^2/((1+x*A(x))*(1+2*x*A(x))) + 3!*x^3/(1-x)^4*A(x)^3/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))) + 4!*x^4/(1-x)^5*A(x)^4/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))*(1+4*x*A(x))) + 5!*x^5/(1-x)^6*A(x)^5/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))*(1+4*x*A(x))*(1+5*x*A(x))) +...

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0, n, x^m/(1-(m+1)*x*A+x*O(x^n))));polcoeff(A, n)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0, n, m!*x^m*A^m/(1-x +x*O(x^n))^(m+1)/prod(k=1, m, 1+k*x*A +x*O(x^n))));polcoeff(, n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A026898, A316367.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 20 2014