A265941 G.f. A(x) satisfies: A(x) = A(A(x)^2) / (x - x^2).

1, -1, 1, -3, 5, -11, 27, -69, 187, -517, 1461, -4163, 11947, -34521, 100261, -292835, 859927, -2538141, 7527545, -22422647, 67055871, -201247141, 605915529, -1829583727, 5539081021, -16810214491, 51130177087, -155839985097, 475899781565, -1455898774263, 4461443964939, -13693120761789, 42089169290257, -129551061180907, 399281615409427

Offset

1,4

Comments

Equals the series reversion of the g.f. of A265940.

Links

Table of n, a(n) for n=1..35.

Formula

Let B(x) be the series reversion of A(x) so that A(B(x)) = x, then

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

(2) B( x*B(x) - x*B(x)^2 ) = x^2.

(3) B(x) = (1 - sqrt(1 - 4*A(x^2)/x)) / 2.

(4) B(x) = C( A(x^2)/x ), where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers A000108.

Example

 G.f.: A(x) = x - x^2 + x^3 - 3*x^4 + 5*x^5 - 11*x^6 + 27*x^7 - 69*x^8 + 187*x^9 - 517*x^10 + 1461*x^11 - 4163*x^12 + 11947*x^13 +...
such that A(A(x)^2) = (x-x^2)*A(x) where
A(A(x)^2) = x^2 - 2*x^3 + 2*x^4 - 4*x^5 + 8*x^6 - 16*x^7 + 38*x^8 - 96*x^9 + 256*x^10 - 704*x^11 + 1978*x^12 - 5624*x^13 + 16110*x^14 +...
Let B(x) be the series reversion of A(x) so that A(B(x)) = x then
B(x) = x + x^2 + x^3 + 3*x^4 + 9*x^5 + 25*x^6 + 71*x^7 + 219*x^8 + 689*x^9 + 2189*x^10 + 7059*x^11 + 23091*x^12 +...+ A265940(n)*x^n +...
such that B( x*B(x) - x*B(x)^2 ) = x^2.

Prog

 (PARI) {a(n) = my(A=x); for(i=1, #binary(n), A = serreverse( (1 - sqrt(1 - 4*subst(A +x*O(x^n), x, x^2)/x) )/2 ) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Crossrefs

Cf. A265940.

Sequence in context: A369344 A204857 A292855 * A333629 A308545 A368808

Adjacent sequences: A265938 A265939 A265940 * A265942 A265943 A265944

Keyword

sign

Author

Paul D. Hanna, Dec 29 2015

Status

approved