A245378 - OEIS

%I #7 Jul 20 2014 05:20:12

%S 1,1,3,10,39,165,743,3507,17199,87126,454159,2430031,13326623,

%T 74856230,430628069,2538270783,15343363603,95233568052,607850790015,

%U 3996223189468,27105153736781,189947851239185,1376864409041417,10330672337146804,80248762443834399,645206035074873681

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

%C Compare g.f. to an identity for C(x) = 1 + x*C(x)^2, the Catalan function:

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

%H Vaclav Kotesovec, <a href="/A245378/b245378.txt">Table of n, a(n) for n = 0..175</a>

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

%e G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 39*x^4 + 165*x^5 + 743*x^6 +...

%e where we have the following identity:

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

%e is equal to

%e A(x) = 1 + x*A(x)*(1+x)/(1+x*A(x)) + 2!*x^2*A(x)^2*(1+x)^2/((1+x*A(x))*(1+2*x*A(x))) + 3!*x^3*A(x)^3*(1+x)^3/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))) + 4!*x^4*A(x)^4*(1+x)^4/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))*(1+4*x*A(x))) + 5!*x^5*A(x)^5*(1+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+x)^(m+1)*(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)^m/prod(k=1, m, 1+k*x*A +x*O(x^n))));polcoeff(A, n)}

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

%Y Cf. A229046, A245373, A245374, A245375, A245376.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 19 2014