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%I #28 Jul 17 2024 05:48:10
%S 1,1,1,3,9,25,71,219,689,2189,7059,23091,76331,254489,855057,2892611,
%T 9843695,33674049,115735093,399448699,1383893277,4811005765,
%U 16777546935,58676543307,205751099671,723221934973,2547846317025,8994482626499,31813908928389,112729816817649,400119890812663,1422409369378787,5064073900824483,18054192195882181,64450109043795285,230358773955266667
%N G.f. A(x) satisfies: A( x*A(x) - x*A(x)^2 ) = x^2.
%C 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
%H Paul D. Hanna, <a href="/A265940/b265940.txt">Table of n, a(n) for n = 1..1025</a>
%F Let B(x) be the series reversion of A(x) so that A(B(x)) = x, then
%F (1) B(x) = B(B(x)^2) / (x - x^2).
%F (2) B(x^2) = x*A(x) - x*A(x)^2.
%F (3) A(x) - A(x)^2 is an odd function.
%F (4) A(x) = (1 - sqrt(1 - 4*B(x^2)/x)) / 2.
%F (5) A(x) = C( B(x^2)/x ), where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers A000108.
%F a(n) ~ c * d^n / n^(3/2), where d = 3.729034131712704991695867178179958688..., c = 0.130930566174680418655094759419136508... . - _Vaclav Kotesovec_, Dec 24 2015
%F 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
%e 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 +...
%e such that A( x*A(x) - x*A(x)^2 ) = x^2.
%e where
%e 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 +...
%e and
%e 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 +...
%e which is an odd function.
%e Compare with B(x), the series reversion of A(x), A(B(x)) = x:
%e 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 +...
%e which satisfies: B(B(x)^2) = (x - x^2)*B(x).
%e SPECIFIC VALUES.
%e A(r) = 1/2 at the radius of convergence r = 0.2681659552257063492958811609250971312812719710081828...
%e where r^2 = A(r/4) and A(-r) = -(sqrt(2) - 1)/2.
%e A(t) = 2/5 at t = 0.255845321447271273745290830537480837403155688844276...
%e A(t) = 1/3 at t = 0.234518570525609093590785779795885030584766907908648...
%e A(t) = 1/4 at t = 0.194622547952562226695813115851351300903854870840228...
%e A(t) = 1/5 at t = 0.164326839348946404126811315954673886155754645645884...
%e A(1/4) = 0.378284164010274536479803372877290855730783167530014...
%e where 1/16 = A( (1/4)*(A(1/4) - A(1/4)^2) ).
%e A(1/5) = 0.259758360807618547135090669720246454745650927887509...
%e where 1/25 = A( (1/5)*(A(1/5) - A(1/5)^2) ).
%e A(1/6) = 0.203614084141603311632304956623078790849219947301354...
%e where 1/36 = A( (1/6)*(A(1/6) - A(1/6)^2) ).
%e A(1/10) = 0.111425303053110288757880516257241040099887886124693...
%e where 1/100 = A( (1/10)*(A(1/10) - A(1/10)^2) ).
%o (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)}
%o for(n=1,40,print1(a(n),", "))
%Y Cf. A271958, A265941, A000108.
%K nonn
%O 1,4
%A _Paul D. Hanna_, Dec 19 2015
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