A182955 G.f. satisfies: A(x) = 1 + x*A(x) * A( x*A(x) )^5.
1, 1, 6, 56, 651, 8671, 126997, 1997798, 33260799, 580270730, 10534337521, 197986746949, 3837397114948, 76473239154148, 1563252546786254, 32716989219013821, 699959257347957763, 15288884723649589585
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 6*x^2 + 56*x^3 + 651*x^4 + 8671*x^5 +...
Related expansions:
A(x*A(x)) = 1 + x + 7*x^2 + 74*x^3 + 953*x^4 + 13846*x^5 +...
A(x*A(x))^5 = 1 + 5*x + 45*x^2 + 520*x^3 + 6950*x^4 + 102481*x^5 +...
The g.f. satisfies:
log(A(x)) = A(x)^5*x + {d/dx x*A(x)^10}*x^2/2! + {d^2/dx^2 x^2*A(x)^15}*x^3/3! + {d^3/dx^3 x^3*A(x)^20}*x^4/4! +...
Programs
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PARI
{a(n)=local(A=1+sum(i=1,n-1,a(i)*x^i+x*O(x^n))); for(i=1,n,A=exp(sum(m=1,n,sum(k=0,n-m,binomial(m+k-1,k)*polcoeff(A^(5*m),k)*x^k)*x^m/m)+x*O(x^n)));polcoeff(A,n)} -
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, 5*k))))}
Formula
G.f. A(x) satisfies:
* A(x) = exp( Sum_{m>=0} {d^m/dx^m x^m*A(x)^(5m+5)} * x^(m+1)/(m+1)! );
* A(x) = exp( Sum_{m>=1} [Sum_{k>=0} C(m+k-1,k)*{[y^k] A(y)^(5m)}*x^k]*x^m/m);
which are equivalent.
Recurrence:
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n with a(0,m)=1, then
a(n,m) = Sum_{k=0..n} m*C(n+m,k)/(n+m) * a(n-k,5k).