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 A271958 G.f. A(x) satisfies: A( x*A(x) - 2*x*A(x)^2 ) = x^2. 2
 1, 2, 6, 32, 178, 1040, 6340, 40000, 258498, 1702640, 11390428, 77184064, 528665660, 3654285120, 25458887576, 178585843712, 1260264395114, 8940927869584, 63732097097436, 456221687622976, 3278343806166228, 23639492652661312, 170998398398401832, 1240505401469155840, 9023083581252148332, 65791560086084269248, 480799272958960336552, 3520985178220566410496, 25834931848716275409960 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 1..300 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 - 2*x^2). (2) B(x^2) = x*A(x) - 2*x*A(x)^2. (3) A(x) - 2*A(x)^2 is an odd function. (4) A(x) = (1 - sqrt(1 - 8*B(x^2)/x)) / 4. (5) A(x) = C( 2*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 = 7.73673466070656248242789... and c = 0.0681159017783868428596... . - Vaclav Kotesovec, Apr 18 2016 The radius of convergence r = 0.12925349567419... = 1/d (d is given above) satisfies r^2 = A(r/8), A(r) = 1/4, A(-r) = -(sqrt(2) - 1)/4. - Paul D. Hanna, Jul 16 2024 EXAMPLE G.f.: A(x) = x + 2*x^2 + 6*x^3 + 32*x^4 + 178*x^5 + 1040*x^6 + 6340*x^7 + 40000*x^8 + 258498*x^9 + 1702640*x^10 + 11390428*x^11 + 77184064*x^12 +... such that A( x*A(x) - 2*x*A(x)^2 ) = x^2 where A(x)^2 = x^2 + 4*x^3 + 16*x^4 + 88*x^5 + 520*x^6 + 3176*x^7 + 20000*x^8 + 129232*x^9 + 851320*x^10 + 5695272*x^11 + 38592032*x^12 +... and A(x) - 2*A(x)^2 = x - 2*x^3 + 2*x^5 - 12*x^7 + 34*x^9 - 116*x^11 + 476*x^13 - 1864*x^15 + 7850*x^17 - 33588*x^19 + 144756*x^21 - 637048*x^23 + 2823468*x^25 +... which is an odd function. Compare with B(x), the series reversion of A(x), A(B(x)) = x: B(x) = x - 2*x^2 + 2*x^3 - 12*x^4 + 34*x^5 - 116*x^6 + 476*x^7 - 1864*x^8 + 7850*x^9 - 33588*x^10 + 144756*x^11 - 637048*x^12 + 2823468*x^13 +... which satisfies: B(B(x)^2) = (x - 2*x^2)*B(x). SPECIFIC VALUES. A(r) = 1/4 where radius convergence r = 0.1292534956741910972081807126589956750525908555... satisfies r^2 = A(r/8) and A(-r) = -(sqrt(2) - 1)/4. A(1/8) = 0.206121210944484449902603598938257342368645158088... where 1/64 = A( (1/8)*(A(1/8) - 2*A(1/8)^2) ). A(1/9) = 0.158893118816327012101085065895684831233779633215... where 1/81 = A( (1/9)*(A(1/9) - 2*A(1/9)^2) ). A(1/10) = 0.133851028600790613492653727895412409356084480579... where 1/100 = A( (1/10)*(A(1/10) - 2*A(1/10)^2) ). PROG (PARI) {a(n) = my(A=x); for(i=1, #binary(n), A = (1 - sqrt(1 - 8*subst(serreverse(A +x*O(x^n)), x, x^2)/x) )/4 ); polcoeff(A, n)} for(n=1, 40, print1(a(n), ", ")) CROSSREFS Cf. A265940. Sequence in context: A108485 A371482 A018940 * A354662 A055596 A318976 Adjacent sequences: A271955 A271956 A271957 * A271959 A271960 A271961 KEYWORD nonn AUTHOR Paul D. Hanna, Apr 17 2016 STATUS approved

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Last modified September 12 11:37 EDT 2024. Contains 375851 sequences. (Running on oeis4.)