A182953 - OEIS

%I #8 Apr 25 2012 10:28:22

%S 1,1,4,25,197,1797,18178,198937,2318858,28487593,366129764,4896068759,

%T 67843403960,971032668429,14319735032679,217136949146091,

%U 3379973833321141,53936100582832901,881318215466710693,14731508761600217914

%N G.f. satisfies: A(x) = 1 + x*A(x) * A( x*A(x) )^3.

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

%F * A(x) = exp( Sum_{m>=0} {d^m/dx^m x^m*A(x)^(3m+3)} * x^(m+1)/(m+1)! );

%F * A(x) = exp( Sum_{m>=1} [Sum_{k>=0} C(m+k-1,k)*{[y^k] A(y)^(3m)}*x^k]*x^m/m);

%F which are equivalent.

%F Recurrence:

%F Let A(x)^m = Sum_{n>=0} a(n,m)*x^n with a(0,m)=1, then

%F a(n,m) = Sum_{k=0..n} m*C(n+m,k)/(n+m) * a(n-k,3k).

%F Given g.f. A(x), then G(x) = 1 + x*A(x)^3 satisfies G(x/G(x)) = 1 + x*G(x)^2 and G(x) is the g.f. of A147664.

%e G.f.: A(x) = 1 + x + 4*x^2 + 25*x^3 + 197*x^4 + 1797*x^5 +...

%e Related expansions:

%e A(x)^3 = 1 + 3*x + 15*x^2 + 100*x^3 + 801*x^4 + 7296*x^5 + 73174*x^6 +...

%e A(x*A(x)) = 1 + x + 5*x^2 + 37*x^3 + 333*x^4 + 3389*x^5 + 37634*x^6 +...

%e A(x*A(x))^3 = 1 + 3*x + 18*x^2 + 142*x^3 + 1311*x^4 + 13461*x^5 +...

%e The g.f. satisfies:

%e log(A(x)) = A(x)^3*x + {d/dx x*A(x)^6}*x^2/2! + {d^2/dx^2 x^2*A(x)^9}*x^3/3! + {d^3/dx^3 x^3*A(x)^12}*x^4/4! +...

%o (PARI) {a(n)=local(A=1+sum(i=1,n-1,a(i)*x^i+x*O(x^n)));

%o for(i=1,n,A=exp(sum(m=1,n,sum(k=0,n-m,binomial(m+k-1,k)*polcoeff(A^(3*m),k)*x^k)*x^m/m)+x*O(x^n)));polcoeff(A,n)}

%o (PARI) {a(n, m=1)=if(n==0, 1, if(m==0, 0^n, sum(k=0, n, m*binomial(n+m, k)/(n+m)*a(n-k, 3*k))))}

%Y Cf. A147664, A030266, A121687, A182954, A182955.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 15 2010