A369547 - OEIS

%I #10 Jan 26 2024 08:33:44

%S 1,1,1,5,11,19,46,150,527,1743,5174,14350,38805,103069,270236,714284,

%T 1943183,5486591,16069552,48586064,150505974,473652950,1502838661,

%U 4778097313,15153189816,47782620920,149511391732,463733630212,1425468348936,4344295289032,13137603866264,39464351087160

%N Expansion of g.f. A(x) satisfying A(x) = A( x^2*(1+x)^4 ) / (x*(1+x)^3).

%C The radius of convergence r of g.f. A(x) solves r*(1+r)^4 = 1 where r = 0.32471795724474602596...

%H Paul D. Hanna, <a href="/A369547/b369547.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)^4 ) / (x*(1+x)^3).

%F (2) R(x*(1+x)^3*A(x)) = x^2*(1+x)^4, where R(A(x)) = x.

%F (3) A(x) = x * Product_{n>=1} F(n), where F(1) = 1+x, and F(n+1) = 1 + (F(n) - 1)^2 * F(n)^4 for n >= 1.

%F (4) A(x)^4 = x^3*B(x) where B(x) is the g.f. of A369554.

%e G.f.: A(x) = x + x^2 + x^3 + 5*x^4 + 11*x^5 + 19*x^6 + 46*x^7 + 150*x^8 + 527*x^9 + 1743*x^10 + 5174*x^11 + 14350*x^12 + ...

%e RELATED SERIES.

%e A(x)^4/x^3 = x + 4*x^2 + 10*x^3 + 36*x^4 + 123*x^5 + 344*x^6 + 976*x^7 + 3000*x^8 + 9505*x^9 + ... + A369554(n)*x^n + ...

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

%e R(x) = x - x^2 + x^3 - 5*x^4 + 15*x^5 - 33*x^6 + 108*x^7 - 417*x^8 + 1228*x^9 - 3767*x^10 + 14409*x^11 - 49801*x^12 + ...

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

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

%e (2) R(x*(1+x)^3*A(x)) = x^2*(1 + x)^4.

%e GENERATING METHOD.

%e Define F(n), a polynomial in x of order 6^(n-1), by the following recurrence:

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

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

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

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

%e F(5) = (1 + x^16 * (1+x)^32 * F(2)^16 * F(3)^8 * F(4)^4),

%e ...

%e F(n+1) = 1 + (F(n) - 1)^2 * F(n)^4

%e ...

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

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

%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)^4 ) - x*(1 + x)^3*F ,#A+1) ); A[n]}

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

%Y Cf. A369554, A369545, A369546, A369548, A369549.

%K nonn

%O 1,4

%A _Paul D. Hanna_, Jan 25 2024