A375445 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^5 )/(1-2*x).
1, 1, 2, 8, 41, 205, 989, 4785, 23881, 124245, 673020, 3771678, 21702164, 127311556, 756930002, 4539680854, 27367146987, 165407567379, 1000581963363, 6051411131431, 36569087782730, 220760294880122, 1331294835476618, 8021165000866546, 48296514171243436, 290695754850732916
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 41*x^4 + 205*x^5 + 989*x^6 + 4785*x^7 + 23881*x^8 + 124245*x^9 + 673020*x^10 + ... where A(x)^2 = A( x^2/(1-2*x)^5 )/(1-2*x). RELATED SERIES. A(x)^2 = 1 + 2*x + 5*x^2 + 20*x^3 + 102*x^4 + 524*x^5 + 2616*x^6 + 13024*x^7 + 66249*x^8 + 348026*x^9 + 1889737*x^10 + ... A(x)^5 = 1 + 5*x + 20*x^2 + 90*x^3 + 470*x^4 + 2566*x^5 + 13885*x^6 + 74435*x^7 + 400530*x^8 + ... + A375455(n+1)*x^n + ... SPECIFIC VALUES. Given the radius of convergence r = 0.15543026888105788743996..., A(r) = 1.4510850920547193207944317544312912656627353873916... where r = (1-2*r)^5 and A(r) = 1/(1-2*r). A(1/7) = 1.273018489928554436323320513425747043274176403249... where A(1/7)^2 = (7/5)*A(343/3125). A(1/8) = 1.198855898496093050319216983995020709132914678012... where A(1/8)^2 = (4/3)*A(16/243). A(1/9) = 1.160774237134743051625929742274648689798420066384... where A(1/9)^2 = (9/7)*A(729/16807). A(1/10) = 1.136139033822992899751347322772302396437733019439... where A(1/10)^2 = (5/4)*A(125/4096).
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..400
Programs
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Mathematica
terms = 26; A[] = 1; Do[A[x] = Sqrt[A[x^2 /(1 - 2x)^5]/(1 - 2x)] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Aug 11 2025 *)
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PARI
{a(n) = my(A=[1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A); A[#A] = (1/2)*polcoeff( subst(Ax, x, x^2/(1-2*x)^5 )/(1-2*x) - Ax^2, #A-1) ); A[n+1]} for(n=0, 30, print1(a(n), ", "))
Formula
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^2 = A( x^2/(1-2*x)^5 )/(1-2*x).
(2) A(x)^4 = A( x^4*y^5 )*y where y = (1-2*x)^3/((1-2*x)^5 - 2*x^2).
(3) A( x^2*(1 + 2*x)^3 ) = A( x/(1+2*x) )^2 / (1+2*x).
The radius of convergence r satisfies r = (1 - 2*r)^5, where A(r) = 1/(1-2*r) and r = 0.1554302688810578874399658483538386517334...
Comments