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A121687 G.f. satisfies A(x) = 1 + x*A(x) * A( x*A(x) )^2. 6
1, 1, 3, 14, 83, 574, 4432, 37244, 335153, 3194510, 32001596, 335019839, 3649450270, 41227610316, 481724831132, 5809341783543, 72177761136925, 922539273876404, 12115001489115910, 163284755614174305 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..150

FORMULA

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

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

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

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

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

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,2k). - Paul D. Hanna, Dec 15 2010

EXAMPLE

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

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

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

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

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

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

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! +...

PROG

(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)}

(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)}

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

(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

CROSSREFS

Cf. variants: A030266, A182953, A182954.

Sequence in context: A215661 A224776 A220910 * A154757 A074535 A256337

Adjacent sequences:  A121684 A121685 A121686 * A121688 A121689 A121690

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 15 2006, Aug 20 2008

STATUS

approved

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Last modified June 24 05:46 EDT 2021. Contains 345416 sequences. (Running on oeis4.)