G.f.: A(x) = x + 2*x^2 + 3*x^3 + 8*x^4 + 15*x^5 + 26*x^6 + 55*x^7 + 124*x^8 + 284*x^9 + 616*x^10 + 1264*x^11 + 2560*x^12 + ...
RELATED SERIES.
(x*A(x))^(1/2) = x + x^2 + x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 13*x^7 + 31*x^8 + 72*x^9 + 142*x^10 + ... + A369545(n)*x^n + ...
Let R(x) be the series reversion of A(x),
R(x) = x - 2*x^2 + 5*x^3 - 18*x^4 + 80*x^5 - 376*x^6 + 1805*x^7 - 8902*x^8 + 45133*x^9 - 233728*x^10 + 1229185*x^11 - 6544420*x^12 + ...
then R(x) and g.f. A(x) satisfy:
(1) R(A(x)) = x,
(2) R(x*A(x)) = x^2*(1 + x)^2.
GENERATING METHOD.
Define F(n), a polynomial in x of order 4^(n-1), by the following recurrence:
F(1) = (1 + x),
F(2) = (1 + x^2 * (1+x)^2),
F(3) = (1 + x^4 * (1+x)^4 * F(2)^2),
F(4) = (1 + x^8 * (1+x)^8 * F(2)^4 * F(3)^2),
F(5) = (1 + x^16 * (1+x)^16 * F(2)^8 * F(3)^4 * F(4)^2),
...
F(n+1) = 1 + (F(n) - 1)^2 * F(n)^2
...
Then the g.f. A(x) equals the infinite product:
A(x) = x * F(1)^2 * F(2)^2 * F(3)^2 * ... * F(n)^2 * ...
that is,
A(x) = x * (1+x)^2 * (1 + x^2*(1+x)^2)^2 * (1 + x^4*(1+x)^4*(1 + x^2*(1+x)^2)^2)^2 * (1 + x^8*(1+x)^8*(1 + x^2*(1+x)^2)^4*(1 + x^4*(1+x)^4*(1 + x^2*(1+x)^2)^2)^2)^2 * ...
SPECIFIC VALUES.
A(t) = 1 at t = 0.3384360046958295823592066275665435383235972422078251618...
A(t) = 4*t at t = 0.3784692870047486765098838556524915548738750059484894725...
A(t) = 9*t at t = 0.4341759819254114048195281285997548356246123884244963574...
A(t) = 16*t at t = 0.4503991198003790196716692640273147965490188133038952185...
A(t) = 25*t at t = 0.4569468453244711249969175826010689125973557341955917137...
A(t) = 36*t at t = 0.4601365544772047206117359824349418391381182470957703685...
A(t) = 49*t at t = 0.4618937559082677697073270302481519549410810789191032971...
A(t) = 64*t at t = 0.4629494015907831262609899780911583211703795156858340575...
A(t) = 81*t at t = 0.4636260570981613757787278132015093203097054838324907566...
A(t) = 100*t at t = 0.464081935314930281442469188416597867797429631824213476...
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