A272463 G.f. A(x) satisfies: A( x^2*A(x) - x^2*A(x)^2 ) = x^3.
1, 1, 2, 4, 12, 36, 112, 360, 1184, 3969, 13506, 46550, 162160, 570076, 2019864, 7205654, 25859788, 93299268, 338207096, 1231194329, 4499137382, 16498152995, 60689045230, 223891151262, 828156420320, 3070760666368, 11411884518800, 42498639965025, 158575098942194, 592761262161240, 2219513277432130, 8323806778346002, 31262959171040784, 117582688976071889, 442819400938052362, 1669735077567533522, 6303424784771599874, 23822450122837267190
Offset: 1
Keywords
Examples
G.f.: A(x) = x + x^2 + 2*x^3 + 4*x^4 + 12*x^5 + 36*x^6 + 112*x^7 + 360*x^8 + 1184*x^9 + 3969*x^10 + 13506*x^11 + 46550*x^12 +... where A( x^2*A(x) - x^2*A(x)^2 ) = x^3. RELATED SERIES. A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 12*x^5 + 36*x^6 + 112*x^7 + 360*x^8 + 1184*x^9 + 3968*x^10 + 13506*x^11 + 46550*x^12 +... A(x) - A(x)^2 = x - x^4 + x^10 - 4*x^13 + 6*x^16 - 27*x^22 + 84*x^25 - 119*x^28 - 70*x^31 + 861*x^34 - 2362*x^37 + 2716*x^40 + 4848*x^43 - 31892*x^46 +... Let B(x) be the series reversion of g.f. A(x), A(B(x)) = x, then B(x) = x - x^2 + x^4 - 4*x^5 + 6*x^6 - 27*x^8 + 84*x^9 - 119*x^10 - 70*x^11 + 861*x^12 - 2362*x^13 + 2716*x^14 + 4848*x^15 - 31892*x^16 +... such that A(x) - A(x)^2 = B(x^3)/x^2. SPECIFIC VALUES. A(1/4) = 0.43750716413214762438474169851025169044... A(1/5) = 0.2728438844373996476937912739143254714680748999753267... A(1/6) = 0.2099915138524924668889213019413855047403227975244006... A(1/7) = 0.1720375832939219643807299314288451907008185008379640... A(1/8) = 0.1461015121522386794470333784876359422356861236711932...
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..500
Programs
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PARI
{a(n) = my(A=[1,1], F=x); for(i=1,n, A=concat(A,0); F=x*Ser(A); A[#A] = polcoeff(x^3 - subst(F,x, x^2*F - x^2*F^2),#A+2) ); A[n]} for(n=1,50,print1(a(n),", "))
Formula
Let B(x) be the series reversion of A(x) so that A(B(x)) = x, then
(1) B(x)^2 = B(B(x)^3) / (x - x^2).
(2) B(x^3)/x^2 = A(x) - A(x)^2.
(3) A(x) = (1 - sqrt(1 - 4*B(x^3)/x^2)) / 2.
(4) A(x) = C( B(x^3)/x^2 ), 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 = 3.9343346144956442477821996521921... and c = 0.1374726251259662065926106275441... . - Vaclav Kotesovec, May 03 2016
Let r be the radius of convergence, then A(r) = 1/2, where r = A(r^2/4)^(1/3) = 0.25417258519791494791315235901614229902947074103519177... = 1/d (d is given above). - Paul D. Hanna, Apr 06 2024