OFFSET
1,2
COMMENTS
Conjecture: [x^(2^n-2^k)] (x/A(x))^(2^n) = 0 for k = 0..n-2 and n >= 2. - Paul D. Hanna, Jan 16 2026
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..521
FORMULA
G.f. A(x) satisfies:
(1) A(x - 2*x^2 - x*G(x^2)) = x, where G(x) = x + (1/2)*(G(x)^2 + G(x^2)) is the g.f. of the Wedderburn-Etherington numbers (A001190).
(2) A(x) = F( sqrt(x*A(x)) ) where F(x)^2 = F( x^2 + 2*F(x)^3 ) and F(x) is the g.f. of A271959.
a(n) ~ c * d^n / n^(3/2), where d = 8.9175668047902516038346068989... and c = 0.056993950617012713508863076... . - Vaclav Kotesovec, May 03 2016
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 9*x^3 + 50*x^4 + 312*x^5 + 2086*x^6 + 14613*x^7 + 105864*x^8 + 786627*x^9 + 5962110*x^10 + 45914544*x^11 + 358247214*x^12 +...
where A(x)^2 = A( (x + 2*A(x)^2)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 22*x^4 + 136*x^5 + 905*x^6 + 6320*x^7 + 45686*x^8 + 338928*x^9 + 2565688*x^10 + 19739244*x^11 + 153893122*x^12 +...
(x + 2*A(x)^2)^2 = x^2 + 4*x^3 + 20*x^4 + 120*x^5 + 784*x^6 + 5412*x^7 + 38808*x^8 + 286200*x^9 + 2156704*x^10 + 16533088*x^11 + 128521172*x^12 +...
sqrt(x*A(x)) = x + x^2 + 4*x^3 + 21*x^4 + 127*x^5 + 832*x^6 + 5746*x^7 + 41191*x^8 + 303602*x^9 + 2286359*x^10 + 17515640*x^11 + 136074960*x^12 + 1069490964*x^13 + 8488634979*x^14 + 67943128844*x^15 + 547784144486*x^16 +...
Let B(x) be the series reversion of g.f. A(x), so that A(B(x)) = x, then
B(x) = x - 2*x^2 - x^3 - x^5 - x^7 - 2*x^9 - 3*x^11 - 6*x^13 - 11*x^15 - 23*x^17 - 46*x^19 - 98*x^21 - 207*x^23 - 451*x^25 +...+ -A001190(n)*x^(2*n+1) +...
such that B(x) = x - 2*x^2 - x*G(x^2), where G(x) = x + (1/2)*(G(x)^2 + G(x^2)).
Let C(x) = series reversion of sqrt(x*A(x)), so that C(x)*A(C(x)) = x^2, then
A(C(x)) = F(x) = x + x^2 + 3*x^3 + 11*x^4 + 46*x^5 + 206*x^6 + 968*x^7 + 4706*x^8 + 23475*x^9 + 119473*x^10 +...+ A271959(n)*x^n +...
where F(x)^2 = F( x^2 + 2*F(x)^3 ).
SPECIFIC VALUES.
A(t) = 1/5 at t = 0.11166610925156917180159981011464702410274916585795...
where 1/25 = A( (t + 2/25)^2 ).
A(t) = 1/6 at t = 0.10634910120197778015196469166129701520042418242100...
A(t) = 1/7 at t = 0.09906460000413080802747512939050107477458403903581...
A(t) = 1/8 at t = 0.09176586532307251809237164312158559250268459839025...
A(1/9) = 0.193716899378316704135816541113258730084910624425207...
where A(1/9)^2 = A( (1/9 + 2*A(1/9)^2)^2 ).
A(1/10) = 0.14546429517891287614768130046564115776024954490631...
where A(1/10)^2 = A( (1/10 + 2*A(1/10)^2)^2 ).
A(1/11) = 0.12312262826542819891757177602341377693863412904549...
where A(1/11)^2 = A( (1/11 + 2*A(1/11)^2)^2 ).
A(1/12) = 0.10787967285688466984977060963500355819842649026285...
where A(1/12)^2 = A( (1/12 + 2*A(1/12)^2)^2 ).
PROG
(PARI) {a(n) = my(A=x+x^2); for(k=1, n, A = truncate(A) + x^3*O(x^k); A = subst(A, x, (x + 2*A^2)^2 )^(1/2); ); polcoef(A, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 23 2016
STATUS
approved
