A369552 - OEIS

%I #15 Jan 26 2024 08:34:05

%S 1,2,3,8,15,26,55,124,284,616,1264,2560,5145,10334,21157,44396,94918,

%T 205404,447798,980176,2147217,4692342,10202201,22035060,47259294,

%U 100704188,213446378,450615024,948696951,1993590770,4184002679,8774184964,18395154470,38578533020,80990279326

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

%C The radius of convergence r of the g.f. A(x) solves r*(1+r)^2 = 1 where r = (((29 + sqrt(837))/2)^(1/3) + ((29 - sqrt(837))/2)^(1/3) - 2)/3 = 0.465571231876768...

%H Paul D. Hanna, <a href="/A369552/b369552.txt">Table of n, a(n) for n = 1..500</a>

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

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

%F (2) R(x*A(x)) = x^2*(1+x)^2, where R(A(x)) = x.

%F (3) A(x) = x * Product_{n>=1} F(n)^2, where F(1) = 1+x, and F(n+1) = 1 + (F(n) - 1)^2 * F(n)^2 for n >= 1.

%e 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 + ...

%e RELATED SERIES.

%e (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 + ...

%e Let R(x) be the series reversion of A(x),

%e 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 + ...

%e then R(x) and g.f. A(x) satisfy:

%e (1) R(A(x)) = x,

%e (2) R(x*A(x)) = x^2*(1 + x)^2.

%e GENERATING METHOD.

%e Define F(n), a polynomial in x of order 4^(n-1), by the following recurrence:

%e F(1) = (1 + x),

%e F(2) = (1 + x^2 * (1+x)^2),

%e F(3) = (1 + x^4 * (1+x)^4 * F(2)^2),

%e F(4) = (1 + x^8 * (1+x)^8 * F(2)^4 * F(3)^2),

%e F(5) = (1 + x^16 * (1+x)^16 * F(2)^8 * F(3)^4 * F(4)^2),

%e ...

%e F(n+1) = 1 + (F(n) - 1)^2 * F(n)^2

%e ...

%e Then the g.f. A(x) equals the infinite product:

%e A(x) = x * F(1)^2 * F(2)^2 * F(3)^2 * ... * F(n)^2 * ...

%e that is,

%e 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 * ...

%e SPECIFIC VALUES.

%e A(t) = 1   at t = 0.3384360046958295823592066275665435383235972422078251618...

%e A(t) = 4*t at t = 0.3784692870047486765098838556524915548738750059484894725...

%e A(t) = 9*t at t = 0.4341759819254114048195281285997548356246123884244963574...

%e A(t) = 16*t at t = 0.4503991198003790196716692640273147965490188133038952185...

%e A(t) = 25*t at t = 0.4569468453244711249969175826010689125973557341955917137...

%e A(t) = 36*t at t = 0.4601365544772047206117359824349418391381182470957703685...

%e A(t) = 49*t at t = 0.4618937559082677697073270302481519549410810789191032971...

%e A(t) = 64*t at t = 0.4629494015907831262609899780911583211703795156858340575...

%e A(t) = 81*t at t = 0.4636260570981613757787278132015093203097054838324907566...

%e A(t) = 100*t at t = 0.464081935314930281442469188416597867797429631824213476...

%o (PARI) {a(n) = my(A=[1],F); for(i=1,n, A=concat(A,0); F=x*Ser(A); A[#A] = polcoeff( subst(F,x, x^2*(1 + x)^2 ) - x*F ,#A+1) ); A[n]}

%o for(n=1,35, print1(a(n),", "))

%Y Cf. A369545, A350432, A369553, A369554, A369555, A369556.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jan 25 2024