A179486 G.f. A(x) satisfies A(x) = x + A(A(x)^3) where A(x) = Sum_{n>=1} a(n)*x^(2*n-1).

1, 1, 3, 12, 56, 285, 1533, 8571, 49311, 290019, 1735845, 10538550, 64741482, 401708636, 2513837931, 15847467276, 100547976684, 641571037002, 4114313992851, 26503239829953, 171416342026944, 1112726829455289

Offset

1,3

Links

Table of n, a(n) for n=1..22.

Formula

G.f. A(x) satisfies: A(x - A(x^3)) = x.

G.f.: A(x) = x + G(x) + G(G(x)) + G(G(G(x))) + G(G(G(G(x)))) +... where G(x) = A(x)^3 = g.f. of A179487.

Given g.f. A(x), define C(x) = A(x^3), then C(x) = x^3 + C(C(x)).

Derivative of g.f. A(x) satisfies: A'(x) = 1/(1 - 3*A(x)^2*A'(A(x)^3)).

Radius of convergence, r, and related values:

r = 0.378590868760339249702289974755919481665219504207137681721365...;

A(r) = 0.5510035125320818261355419567786182869427265480378585343298... where r = A(r) - A(A(r)^3);

A(r)^3 = 0.1672873502451522851544780724841939477291722823741494215...;

A(A(r)^3) = 0.1724126437717425764332519820226988052775070438307208...;

A'(A(r)^3) = 1.0979182660346808662695442970765885990300854399844658... where A'(A(r)^3) = 1/(3*A(r)^2);

Limit a(n+1)/a(n) = 1/r^2 = 6.9768555281242291444841704586123374638...

Let V(x) = x/(x - A(x^3)) then V'(A(r)) = 1/r, V(z) - z*V'(z) = 0 at z=A(r), and V(A(x)) = A(x)/x for all x.

Example

G.f.: A(x) = x + x^3 + 3*x^5 + 12*x^7 + 56*x^9 + 285*x^11 +...
A(x)^3 = x^3 + 3*x^5 + 12*x^7 + 55*x^9 + 276*x^11 + 1470*x^13 +...
The series reversion of A(x) equals x - A(x^3), therefore
x = A(x - x^3 - x^9 - 3*x^15 - 12*x^21 - 56*x^27 - 285*x^33 -...).
Let G(x) = A(x)^3 be the g.f. of A179487, then
G(G(x)) = x^9 + 9*x^11 + 63*x^13 + 411*x^15 + 2619*x^17 +...,
G(G(G(x))) = x^27 + 27*x^29 + 432*x^31 + 5364*x^33 +..., and
G(G(G(G(x)))) = x^81 + 81*x^83 + 3483*x^85 +...
where A(x) = x + G(x) + G(G(x)) + G(G(G(x))) + G(G(G(G(x)))) +...

Prog

(PARI) {a(n)=local(A=x+x^3); for(i=0, n, A=serreverse(x-subst(A, x, x^3+x^2*O(x^(2*n))))); polcoeff(A, 2*n-1)}
(Maxima)
Co(n, k, F):=if k=1 then F(n) else sum(F(i+1)*Co(n-i-1, k-1, F), i, 0, n-k);
b(n):=if n=1 then 1 else sum(if 3*k>n then 0 else Co(n, 3*k, b)*b(k), k, 1, n);
a(n):=b(2*n-1);
makelist(a(n), n, 1, 7); [Vladimir Kruchinin, Jun 28 2011]
(Maxima)
T(n, m):=if n=m then 1 else kron_delta(n, m)+sum(binomial(m, j)*sum(if 3*k<=n-j then T(n-j, 3*k)*T(k, m-j) else 0, k, m-j, n-j), j, 0, m-1);
makelist(T(n, 1), n, 1, 12); [Vladimir Kruchinin, May 02 2012]

Crossrefs

Cf. A179487, A141200 (variant).

Sequence in context: A195261 A276902 A192132 * A369482 A366097 A074533

Adjacent sequences: A179483 A179484 A179485 * A179487 A179488 A179489

Keyword

nonn

Author

Paul D. Hanna, Aug 12 2010

Extensions

Typo in example corrected by Paul D. Hanna, Aug 13 2010

Status

approved