A229808 G.f. A(x) satisfies: A(x) = x + 2*B(x)*C(x), where B(x) = x + 3*A(x)*C(x) and C(x) = x + 5*A(x)*B(x).

1, 2, 16, 122, 1096, 10322, 102856, 1054322, 11104336, 119175842, 1299844576, 14360860202, 160403229016, 1808192266322, 20545886851096, 235069897341122, 2705792876561056, 31312112470672322, 364079737402597936, 4251402314343575642, 49835252820257276776

Offset

1,2

Links

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

Formula

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

(1) A = x + 2*x^2*(1+3*A)*(1+5*A)/(1-15*A^2)^2.

(2) A = x*(1+2*B)/(1-10*B^2) where B = x*(1+3*A)/(1-15*A^2) is the g.f. of A229809.

(3) A = x*(1+2*C)/(1-6*C^2) where C = x*(1+5*A)/(1-15*A^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.: A(x) = x + 2*x^2 + 16*x^3 + 122*x^4 + 1096*x^5 + 10322*x^6 +...
Related series:
B(x) = x + 3*x^2 + 21*x^3 + 153*x^4 + 1401*x^5 + 13083*x^6 +...
C(x) = x + 5*x^2 + 25*x^3 + 215*x^4 + 1825*x^5 + 17525*x^6 +...
where A(x) = x + 2*B(x)*C(x).
(A(x)^2 - x*A(x))/2 = 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(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=x); for(i=1, n, A=x+2*x^2*(1+3*A)*(1+5*A)/(1-15*A^2 +x*O(x^n))^2); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A229809 (B(x)), A229810 (C(x)).

Sequence in context: A200820 A209361 A341925 * A301947 A027309 A241466

Adjacent sequences: A229805 A229806 A229807 * A229809 A229810 A229811

Keyword

nonn

Author

Paul D. Hanna, Sep 30 2013

Status

approved