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 A265940 G.f. A(x) satisfies: A( x*A(x) - x*A(x)^2 ) = x^2. 8
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 17 04:45 EDT 2024. Contains 375985 sequences. (Running on oeis4.)