A258315 G.f. C(x) satisfies: C(x) = 1 + 2*x*A(x)*B(x) where A(x) = B(x)*C(x) and B(x) = 1 + x*A(x)*C(x).

1, 2, 8, 46, 304, 2178, 16456, 129086, 1041248, 8582274, 71964232, 611954286, 5264786448, 45741886786, 400776143752, 3537136653566, 31417018218688, 280616550025218, 2518975669228936, 22712641808517166, 205612543320237808, 1868112977079278594, 17028815533533595080

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Formula

G.f. C(x) satisfies:

(1) C(x) = 1 + 2*x*C(x)*(1 - C(x) + C(x)^2) + x^2*C(x)^4*(1 - C(x)).

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

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

Other relations involving A=A(x), B=B(x), and C=C(x) are:

(a) B = (1 + x*A) / (1 - 2*x^2*A^2).

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

(c) B = 1/(1 - x*C^2).

(d) C = 1/(1 - 2*x*B^2).

Example

G.f.: C(x) = 1 + 2*x + 8*x^2 + 46*x^3 + 304*x^4 + 2178*x^5 + 16456*x^6 +...
where C(x) = 1 + 2*x*A(x)*B(x):
A(x) = 1 + 3*x + 15*x^2 + 93*x^3 + 641*x^4 + 4719*x^5 + 36335*x^6 +...
B(x) = 1 + x + 5*x^2 + 29*x^3 + 193*x^4 + 1389*x^5 + 10525*x^6 +...
Related series:
A(x)*B(x) = 1 + 4*x + 23*x^2 + 152*x^3 + 1089*x^4 + 8228*x^5 +...
A(x)*C(x) = 1 + 5*x + 29*x^2 + 193*x^3 + 1389*x^4 + 10525*x^5 +...

Prog

(PARI) {a(n)=local(A=1+x, B=1+x, C=1+2*x); for(i=1, n, A = B*C +x*O(x^n); B = 1 + x*A*C + x*O(x^n); C = 1 + 2*x*A*B + x*O(x^n)); polcoeff(C, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(C=1); C = (1/x)*serreverse( x^2/(x + 2*serreverse( x*(1-2*x^2)/(1+x +x*O(x^n)) )^2) ); polcoeff(C, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A258313 (A(x)), A258314 (B(x)).

Sequence in context: A269006 A266507 A202081 * A334498 A006664 A276367

Adjacent sequences: A258312 A258313 A258314 * A258316 A258317 A258318

Keyword

nonn

Author

Paul D. Hanna, May 25 2015

Status

approved