A317999 G.f. A(x) satisfies: Sum_{n>=1} (-1)^n * (A(x) - (-1)^n*A(-x))^n = 0.

1, 2, 4, 24, 112, 608, 3392, 19456, 114688, 681984, 4120576, 25182208, 155394048, 967598080, 6070190080, 38322601984, 243289358336, 1552850223104, 9960145289216, 64109305921536, 413780210089984, 2683663674245120, 17513860521000960, 114061957027332096, 735229028562108416, 4844244732571811840, 33982887942858735616, 218245067017509928960, 906594523232033832960

Offset

1,2

Links

Table of n, a(n) for n=1..29.

Formula

G.f. A(x) satisfies:

(1) A(-A(-x)) = x.

(2a) 0 = Sum_{n>=1} (-1)^n * (A(x) - (-1)^n*A(-x))^n.

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

(3) 0 = x*(1-x)*(1+x)^2 - (1-x)^2*A(A(x)) + (1 + x + 2*x^2)*A(A(x))^2 + A(A(x))^3 - A(A(x))^4.

(4) A(A(x)) = G(x) such that Sum_{n>=1} (x + (-1)^n*G(x))^n = 0, where G(x) is the g.f. of A317998.

Example

G.f. A(x) = x + 2*x^2 + 4*x^3 + 24*x^4 + 112*x^5 + 608*x^6 + 3392*x^7 + 19456*x^8 + 114688*x^9 + 681984*x^10 + 4120576*x^11 + 25182208*x^12 + ...
such that
0 = (A(x) + A(-x)) - (A(x) - A(-x))^2 + (A(x) + A(-x))^3 - (A(x) - A(-x))^4 + (A(x) + A(-x))^5 - (A(x) - A(-x))^6 + (A(x) + A(-x))^7 - (A(x) - A(-x))^8 + ...
Also,
0 = (x - A(A(x))) + (x + A(A(x)))^2 + (x - A(A(x)))^3 + (x + A(A(x)))^4 + (x - A(A(x)))^5 + (x + A(A(x)))^6 + (x - A(A(x)))^7 + (x + A(A(x)))^8 + ...
RELATED SERIES.
A(A(x)) = x + 4*x^2 + 16*x^3 + 96*x^4 + 640*x^5 + 4480*x^6 + 32768*x^7 + 247552*x^8 + 1915904*x^9 + 15113216*x^10 + ... + A317998(n)*x^n + ...

Crossrefs

Cf. A317998.

Sequence in context: A192382 A232205 A170931 * A371892 A164313 A087981

Adjacent sequences: A317996 A317997 A317998 * A318000 A318001 A318002

Keyword

nonn

Author

Paul D. Hanna, Aug 21 2018

Status

approved