A229810 G.f. C(x) satisfies: C(x) = x + 5*A(x)*B(x), where A(x) = x + 2*B(x)*C(x) and B(x) = x + 3*A(x)*C(x).

1, 5, 25, 215, 1825, 17525, 172525, 1772435, 18615025, 199711445, 2176008625, 24027883055, 268226469025, 3022357427765, 34328716158325, 392633368190075, 4518132270765025, 52271679549480485, 607648547991054025, 7094152934668535495, 83143099009577766625

Offset

1,2

Links

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

Formula

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

(1) C = x + 5*x^2*(1+2*C)*(1+3*C)/(1-6*C^2)^2.

(2) C = x*(1+5*A)/(1-15*A^2) where A = x*(1+2*C)/(1-6*C^2) is the g.f. of A229808.

(3) C = x*(1+5*B)/(1-10*B^2) where B = x*(1+3*C)/(1-6*C^2) is the g.f. of A229809.

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

Crossrefs

Cf. A229808 (A(x)), A229809 (B(x)).

Sequence in context: A254335 A193939 A352075 * A080631 A080632 A245166

Adjacent sequences: A229807 A229808 A229809 * A229811 A229812 A229813

Keyword

nonn

Author

Paul D. Hanna, Sep 30 2013

Status

approved