A258314 G.f. B(x) satisfies: B(x) = 1 + x*A(x)*C(x) where A(x) = B(x)*C(x) and C(x) = 1 + 2*x*A(x)*B(x).

1, 1, 5, 29, 193, 1389, 10525, 82729, 668321, 5514873, 46285861, 393889605, 3390819777, 29475696197, 258371636989, 2281190100625, 20268424498753, 181090741905393, 1625999443354501, 14664436054418477, 132781726001339713, 1206629736349162909, 11000943167309740701, 100596316305081808185

Offset

0,3

Links

Table of n, a(n) for n=0..23.

Formula

G.f. B(x) satisfies:

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

(2) B(x) = sqrt( (1/x)*Series_Reversion( x*(1-2*x)^2 / (1-x + x^2*G(-x^2))^2 ) ), where G(x) = 1 + x*G(x)^2 is the g.f. of the Catalan numbers.

(3) x = ( sqrt(1 - 8*B(x) + 8*B(x)^2) - (1 - 2*B(x))^2 ) / (8*B(x)^3*(1-B(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.: B(x) = 1 + x + 5*x^2 + 29*x^3 + 193*x^4 + 1389*x^5 + 10525*x^6 +...
where B(x) = 1 + x*A(x)*C(x):
A(x) = 1 + 3*x + 15*x^2 + 93*x^3 + 641*x^4 + 4719*x^5 + 36335*x^6 +...
C(x) = 1 + 2*x + 8*x^2 + 46*x^3 + 304*x^4 + 2178*x^5 + 16456*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(B, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(B=1); B = sqrt( (1/x)*serreverse( x*(1-2*x)^2 / (1-x + x*serreverse(x/(1-x^2 +x*O(x^n))))^2 ) ); polcoeff(B, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A258313 (A(x)), A258315 (C(x)).

Sequence in context: A062191 A367240 A171267 * A225030 A188143 A095000

Adjacent sequences: A258311 A258312 A258313 * A258315 A258316 A258317

Keyword

nonn

Author

Paul D. Hanna, May 25 2015

Status

approved