A121687 - OEIS

%I #21 Feb 13 2015 16:49:16

%S 1,1,3,14,83,574,4432,37244,335153,3194510,32001596,335019839,

%T 3649450270,41227610316,481724831132,5809341783543,72177761136925,

%U 922539273876404,12115001489115910,163284755614174305

%N G.f. satisfies A(x) = 1 + x*A(x) * A( x*A(x) )^2.

%H Vaclav Kotesovec, <a href="/A121687/b121687.txt">Table of n, a(n) for n = 0..150</a>

%F G.f. satisfies G(x) = x/(1 + x*A(x)^2) where G(x*A(x)) = x.

%F G.f. satisfies A(x) = 1/(1 - x*A(x*A(x))^2).

%F G.f. satisfies the following two equations (which are equivalent)

%F A(x) = exp( Sum_{m>=0} {d^m/dx^m x^m*A(x)^(2m+2)} * x^(m+1)/(m+1)! )

%F A(x) = exp( Sum_{m>=1} [Sum_{k>=0} C(m+k-1,k)*{[y^k] A(y)^(2m)}*x^k]*x^m/m). - _Paul D. Hanna_, Dec 15 2010

%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,2k). - _Paul D. Hanna_, Dec 15 2010

%e G.f. A(x) = 1 + x + 3*x^2 + 14*x^3 + 83*x^4 + 574*x^5 + 4432*x^6 +...

%e A(x)^2 = 1 + 2*x + 7*x^2 + 34*x^3 + 203*x^4 + 1398*x^5 + 10706*x^6 +...

%e A(x*A(x)) = 1 + x + 4*x^2 + 23*x^3 + 160*x^4 + 1259*x^5 + 10833*x^6 +...

%e A(x*A(x))^2 = 1 + 2*x + 9*x^2 + 54*x^3 + 382*x^4 + 3022*x^5 + 25993*x^6 +...

%e A(x)*A(x*A(x))^2 = 1 + 3*x + 14*x^2 + 83*x^3 + 574*x^4 + 4432*x^5 +...

%e The logarithm of the g.f. is given by:

%e log(A(x)) = A(x)^2*x + {d/dx x*A(x)^4}*x^2/2! + {d^2/dx^2 x^2*A(x)^6}*x^3/3! + {d^3/dx^3 x^3*A(x)^8}*x^4/4! +...

%o (PARI) {a(n)=local(A=1+x);for(i=0,n,A=serreverse(x/(1+x*(A +x*O(x^n))^2))/x); polcoeff(A,n)}

%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1/(1-x*subst(A^2,x,x*A)));polcoeff(A,n)}

%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^(2*m),k)*x^k)*x^m/m)+x*O(x^n)));polcoeff(A,n)} \\ _Paul D. Hanna_, Dec 15 2010

%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))))} \\ _Paul D. Hanna_, Dec 15 2010

%Y Cf. variants: A030266, A182953, A182954.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 15 2006, Aug 20 2008