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Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^5 )/(1-2*x).
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%I #9 Jan 16 2025 17:47:26

%S 1,1,2,8,41,205,989,4785,23881,124245,673020,3771678,21702164,

%T 127311556,756930002,4539680854,27367146987,165407567379,

%U 1000581963363,6051411131431,36569087782730,220760294880122,1331294835476618,8021165000866546,48296514171243436,290695754850732916

%N Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^5 )/(1-2*x).

%C Compare to M(x)^2 = M( x^2/(1-2*x) )/(1-2*x), where M(x) = 1 + x*M(x) + x^2*M(x)^2 is the g.f. of the Motzkin numbers (A001006).

%C Compare to C(x)^2 = C( x^2/(1-2*x)^2 )/(1-2*x), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

%H Paul D. Hanna, <a href="/A375445/b375445.txt">Table of n, a(n) for n = 0..400</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

%F (1) A(x)^2 = A( x^2/(1-2*x)^5 )/(1-2*x).

%F (2) A(x)^4 = A( x^4*y^5 )*y where y = (1-2*x)^3/((1-2*x)^5 - 2*x^2).

%F (3) A( x^2*(1 + 2*x)^3 ) = A( x/(1+2*x) )^2 / (1+2*x).

%F The radius of convergence r satisfies r = (1 - 2*r)^5, where A(r) = 1/(1-2*r) and r = 0.1554302688810578874399658483538386517334...

%e 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 + ...

%e where A(x)^2 = A( x^2/(1-2*x)^5 )/(1-2*x).

%e RELATED SERIES.

%e 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 + ...

%e 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 + ...

%e SPECIFIC VALUES.

%e Given the radius of convergence r = 0.15543026888105788743996...,

%e A(r) = 1.4510850920547193207944317544312912656627353873916...

%e where r = (1-2*r)^5 and A(r) = 1/(1-2*r).

%e A(1/7) = 1.273018489928554436323320513425747043274176403249...

%e where A(1/7)^2 = (7/5)*A(343/3125).

%e A(1/8) = 1.198855898496093050319216983995020709132914678012...

%e where A(1/8)^2 = (4/3)*A(16/243).

%e A(1/9) = 1.160774237134743051625929742274648689798420066384...

%e where A(1/9)^2 = (9/7)*A(729/16807).

%e A(1/10) = 1.136139033822992899751347322772302396437733019439...

%e where A(1/10)^2 = (5/4)*A(125/4096).

%o (PARI) {a(n) = my(A=[1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);

%o A[#A] = (1/2)*polcoeff( subst(Ax, x, x^2/(1-2*x)^5 )/(1-2*x) - Ax^2, #A-1) ); A[n+1]}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A001006, A000108, A375455, A375443, A375444.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 19 2024