A182953 G.f. satisfies: A(x) = 1 + x*A(x) * A( x*A(x) )^3.

1, 1, 4, 25, 197, 1797, 18178, 198937, 2318858, 28487593, 366129764, 4896068759, 67843403960, 971032668429, 14319735032679, 217136949146091, 3379973833321141, 53936100582832901, 881318215466710693, 14731508761600217914

Offset

0,3

Links

Table of n, a(n) for n=0..19.

Formula

G.f. A(x) satisfies:

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

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

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.

Example

G.f.: A(x) = 1 + x + 4*x^2 + 25*x^3 + 197*x^4 + 1797*x^5 +...
Related expansions:
A(x)^3 = 1 + 3*x + 15*x^2 + 100*x^3 + 801*x^4 + 7296*x^5 + 73174*x^6 +...
A(x*A(x)) = 1 + x + 5*x^2 + 37*x^3 + 333*x^4 + 3389*x^5 + 37634*x^6 +...
A(x*A(x))^3 = 1 + 3*x + 18*x^2 + 142*x^3 + 1311*x^4 + 13461*x^5 +...
The g.f. satisfies:
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! +...

Prog

(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^(3*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, 3*k))))}

Crossrefs

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

Sequence in context: A202827 A065735 A212694 * A327074 A337167 A140094

Adjacent sequences: A182950 A182951 A182952 * A182954 A182955 A182956

Keyword

nonn

Author

Paul D. Hanna, Dec 15 2010

Status

approved