A191557 G.f. satisfies: A(A(x))^2 = x^2 + 4*A(x)^3.

1, 1, 1, -1, -5, 6, 57, -68, -996, 1151, 23487, -26316, -703858, 769268, 25912425, -27791388, -1146924362, 1212941187, 60112150656, -62911402588, -3686975047595, 3828485422340, 262043300715095, -270475215554448, -21394371719691000

Offset

1,5

Comments

Compare the g.f. to the following property of G(x) = x*sqrt(1+4*x):

G(G(x))^2 = x^2 + 4*x^3 + 4*G(x)^3.

Links

Table of n, a(n) for n=1..25.

Formula

G.f. satisfies: A(x)^2 = A(-x)^2 + 4*x^3.

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

G.f. satisfies: A(x) = B(x) + x^3/B(x) where B(x) = (A(x) - A(-x))/2.

Example

G.f.: A(x) = x + x^2 + x^3 - x^4 - 5*x^5 + 6*x^6 + 57*x^7 - 68*x^8 +...
Note that x^3 is the only odd power of x in A(x)^2:
A(x)^2 = x^2 + 2*x^3 + 3*x^4 - 11*x^6 + 117*x^8 - 2001*x^10 +...
Illustrate A(A(x))^2 = x^2 + 4*A(x)^3 by the expansions:
A(A(x))^2 = x^2 + 4*x^3 + 12*x^4 + 24*x^5 + 16*x^6 - 60*x^7 - 72*x^8 + 640*x^9 + 768*x^10 - 11160*x^11 - 12916*x^12 +...
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 4*x^6 - 15*x^7 - 18*x^8 + 160*x^9 + 192*x^10 - 2790*x^11 - 3229*x^12 +...
G.f. of odd bisection B(x) = (A(x) - A(-x))/2 begins:
B(x) = x + x^3 - 5*x^5 + 57*x^7 - 996*x^9 + 23487*x^11 +...
where A(x) = B(x) + x^3/B(x).

Prog

(PARI) {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=A-(subst(A, x, A)-x*sqrt(1+4*A^3/x^2))/2); polcoeff(A, n)}

Crossrefs

Cf. A107700, A191565, A138740.

Sequence in context: A157805 A256291 A299243 * A223530 A132444 A111504

Adjacent sequences: A191554 A191555 A191556 * A191558 A191559 A191560

Keyword

sign

Author

Paul D. Hanna, Jun 06 2011

Status

approved