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A375445
Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^5 )/(1-2*x).
3
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
0,3
COMMENTS
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).
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).
LINKS
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...
EXAMPLE
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).
PROG
(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), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 19 2024
STATUS
approved