A229813 G.f. C(x) satisfies: C(x) = x + 3*A(x)*B(x), where A(x) = x + B(x)*C(x) and B(x) = x + 2*A(x)*C(x).

1, 3, 9, 45, 225, 1275, 7389, 44745, 276849, 1750275, 11236833, 73114437, 480936033, 3193267467, 21372274341, 144040951953, 976706321121, 6658535367555, 45611307797049, 313782691341597, 2167022784185505, 15018193080454491, 104413014897103917, 728039790269173209

Offset

1,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..300

Vaclav Kotesovec, Recurrence (of order 9)

Formula

G.f. C = C(x) satisfies:

(1) C = x + 3*x^2*(1+C)*(1+2*C)/(1-2*C^2)^2.

(2) C = x*(1+3*A)/(1-6*A^2) where A = x*(1+C)/(1-2*C^2) is the g.f. of A229811.

(3) C = x*(1+3*B)/(1-3*B^2) where B = x*(1+2*C)/(1-2*C^2) is the g.f. of A229812.

The g.f.s A = A(x) (A229811), B = B(x) (A229812), C = C(x) (A229813), satisfy:

A*B*C = (A^2 - x*A) = (B^2 - x*B)/2 = (C^2 - x*C)/3.

a(n) ~ c*d^n/n^(3/2), where d = 7.438049365405038364... is the root of the equation -9 - 114*d - 442*d^2 - 792*d^3 - 660*d^4 - 432*d^5 - 192*d^6 - 24*d^7 + 8*d^8 = 0 and c = 0.102311163701744278796886833630056159781... - Vaclav Kotesovec, Sep 30 2013

Example

G.f.: C(x) = x + 3*x^2 + 9*x^3 + 45*x^4 + 225*x^5 + 1275*x^6 + 7389*x^7 +...
Related series:
A(x) = x + x^2 + 5*x^3 + 23*x^4 + 121*x^5 + 673*x^6 + 3953*x^7 +...
B(x) = x + 2*x^2 + 8*x^3 + 34*x^4 + 184*x^5 + 1010*x^6 + 5936*x^7 +...
where C(x) = x + 3*A(x)*B(x).
(C(x)^2 - x*C(x))/3 = A(x)*B(x)*C(x) = x^3 + 6*x^4 + 33*x^5 + 192*x^6 + 1145*x^7 + 7038*x^8 + 44093*x^9 + 281232*x^10 + 1818513*x^11 + 11899830*x^12 +...

Prog

(PARI) {a(n)=local(A=x+x^2, B=x+2*x^2, C=x+3*x^2); for(i=1, n, A=x+B*C+x*O(x^n); B=x+2*A*C+x*O(x^n); C=x+3*A*B+x*O(x^n)); polcoeff(C, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n)=local(C=x); for(i=1, n, C=x+3*x^2*(1+C)*(1+2*C)/(1-2*C^2 +x*O(x^n))^2); polcoeff(C, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A229811 (A(x)), A229812 (B(x)).

Sequence in context: A327648 A262129 A012821 * A262130 A262131 A262132

Adjacent sequences: A229810 A229811 A229812 * A229814 A229815 A229816

Keyword

nonn

Author

Paul D. Hanna, Sep 30 2013

Status

approved