A265940 G.f. A(x) satisfies: A( x*A(x) - x*A(x)^2 ) = x^2.

1, 1, 1, 3, 9, 25, 71, 219, 689, 2189, 7059, 23091, 76331, 254489, 855057, 2892611, 9843695, 33674049, 115735093, 399448699, 1383893277, 4811005765, 16777546935, 58676543307, 205751099671, 723221934973, 2547846317025, 8994482626499, 31813908928389, 112729816817649, 400119890812663, 1422409369378787, 5064073900824483, 18054192195882181, 64450109043795285, 230358773955266667

Offset

1,4

Comments

Compare to C( x*C(x) + x*C(x)^2 ) = C(x)^2 where C(x) = x + C(x)^2 is the g.f. of the Catalan numbers (A000108). - Paul D. Hanna, Jul 16 2024

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1025

Formula

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

(1) B(x) = B(B(x)^2) / (x - x^2).

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

(3) A(x) - A(x)^2 is an odd function.

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

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

a(n) ~ c * d^n / n^(3/2), where d = 3.729034131712704991695867178179958688..., c = 0.130930566174680418655094759419136508... . - Vaclav Kotesovec, Dec 24 2015

The radius of convergence r = 0.2681659552257... = 1/d (d is given above) satisfies r^2 = A(r/4), A(r) = 1/2, and A(-r) = -(sqrt(2) - 1)/2. - Paul D. Hanna, Jul 16 2024

Example

G.f.: A(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 +...
such that A( x*A(x) - x*A(x)^2 ) = x^2.
where
A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 8*x^5 + 25*x^6 + 74*x^7 + 219*x^8 + 684*x^9 + 2189*x^10 + 7070*x^11 + 23091*x^12 + 76304*x^13 +...
and
A(x) - A(x)^2 = x - x^3 + x^5 - 3*x^7 + 5*x^9 - 11*x^11 + 27*x^13 - 69*x^15 + 187*x^17 - 517*x^19 + 1461*x^21 - 4163*x^23 + 11947*x^25 +...
which is an odd function.
Compare with B(x), the series reversion of A(x), A(B(x)) = x:
B(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 +...+ A265941(n)*x^n +...
which satisfies: B(B(x)^2) = (x - x^2)*B(x).
SPECIFIC VALUES.
A(r) = 1/2 at the radius of convergence r = 0.2681659552257063492958811609250971312812719710081828...
where r^2 = A(r/4) and A(-r) = -(sqrt(2) - 1)/2.
A(t) = 2/5 at t = 0.255845321447271273745290830537480837403155688844276...
A(t) = 1/3 at t = 0.234518570525609093590785779795885030584766907908648...
A(t) = 1/4 at t = 0.194622547952562226695813115851351300903854870840228...
A(t) = 1/5 at t = 0.164326839348946404126811315954673886155754645645884...
A(1/4) = 0.378284164010274536479803372877290855730783167530014...
where 1/16 = A( (1/4)*(A(1/4) - A(1/4)^2) ).
A(1/5) = 0.259758360807618547135090669720246454745650927887509...
where 1/25 = A( (1/5)*(A(1/5) - A(1/5)^2) ).
A(1/6) = 0.203614084141603311632304956623078790849219947301354...
where 1/36 = A( (1/6)*(A(1/6) - A(1/6)^2) ).
A(1/10) = 0.111425303053110288757880516257241040099887886124693...
where 1/100 = A( (1/10)*(A(1/10) - A(1/10)^2) ).

Prog

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

Crossrefs

Cf. A271958, A265941, A000108.

Sequence in context: A211292 A291683 A079857 * A309104 A211282 A211298

Adjacent sequences: A265937 A265938 A265939 * A265941 A265942 A265943

Keyword

nonn

Author

Paul D. Hanna, Dec 19 2015

Status

approved