A229809 G.f. B(x) satisfies: B(x) = x + 3*A(x)*C(x), where A(x) = x + 2*B(x)*C(x) and C(x) = x + 5*A(x)*B(x).

1, 3, 21, 153, 1401, 13083, 130401, 1334613, 14048481, 150675363, 1642689501, 18141639153, 202569926121, 2282923197003, 25934267869401, 296661689600733, 3414170774033601, 39503727765284163, 459266320623014181, 5362271816243579913, 62850255058362431001

Offset

1,2

Links

Table of n, a(n) for n=1..21.

Formula

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

(1) B = x + 3*x^2*(1+2*B)*(1+5*B)/(1-10*B^2)^2.

(2) B = x*(1+3*A)/(1-15*A^2) where A = x*(1+2*B)/(1-10*B^2) is the g.f. of A229808.

(3) B = x*(1+3*C)/(1-6*C^2) where C = x*(1+5*B)/(1-10*B^2) is the g.f. of A229810.

The g.f.s A = A(x) (A229808), B = B(x) (A229809), C = C(x) (A229810), satisfy:

A*B*C = (A^2 - x*A)/2 = (B^2 - x*B)/3 = (C^2 - x*C)/5.

Example

G.f.: B(x) = x + 3*x^2 + 21*x^3 + 153*x^4 + 1401*x^5 + 13083*x^6 +...
Related series:
A(x) = x + 2*x^2 + 16*x^3 + 122*x^4 + 1096*x^5 + 10322*x^6 +...
C(x) = x + 5*x^2 + 25*x^3 + 215*x^4 + 1825*x^5 + 17525*x^6 +...
where B(x) = x + 3*A(x)*C(x).
(B(x)^2 - x*B(x))/3 = x^3 + 10*x^4 + 93*x^5 + 920*x^6 + 9305*x^7 + 97050*x^8 + 1031737*x^9 +...

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+2*B*C+x*O(x^n); B=x+3*A*C+x*O(x^n); C=x+5*A*B+x*O(x^n)); polcoeff(B, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n)=local(B=x); for(i=1, n, B=x+3*x^2*(1+2*B)*(1+5*B)/(1-10*B^2 +x*O(x^n))^2); polcoeff(B, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A229808 (A(x)), A229810 (C(x)).

Sequence in context: A183412 A155627 A163472 * A074575 A091171 A226560

Adjacent sequences: A229806 A229807 A229808 * A229810 A229811 A229812

Keyword

nonn

Author

Paul D. Hanna, Sep 30 2013

Status

approved