A369548 - OEIS

%I #10 Jan 26 2024 08:33:48

%S 1,1,1,6,16,31,76,267,1067,4158,14773,48624,154124,477235,1444512,

%T 4319533,13001635,39984251,126648990,414577918,1401344342,4867242685,

%U 17246577931,61892834305,223556783337,808761979971,2919883601666,10493659868180,37480931951560,132938081422654

%N Expansion of g.f. A(x) satisfying A(x) = A( x^2*(1+x)^5 ) / (x*(1+x)^4).

%C The radius of convergence r of g.f. A(x) solves r*(1+r)^5 = 1 where r = 0.2851990332453493679...

%H Paul D. Hanna, <a href="/A369548/b369548.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)^5 ) / (x*(1+x)^4).

%F (2) R(x*(1+x)^4*A(x)) = x^2*(1+x)^5, 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)^5 for n >= 1.

%F (4) A(x)^5 = x^4*B(x) where B(x) is the g.f. of A369555.

%e G.f.: A(x) = x + x^2 + x^3 + 6*x^4 + 16*x^5 + 31*x^6 + 76*x^7 + 267*x^8 + 1067*x^9 + 4158*x^10 + 14773*x^11 + 48624*x^12 + ...

%e RELATED SERIES.

%e A(x)^5/x^4 = x + 5*x^2 + 15*x^3 + 60*x^4 + 245*x^5 + 826*x^6 + 2685*x^7 + 9285*x^8 + 33170*x^9 + ... + A369555(n)*x^n + ...

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

%e R(x) = x - x^2 + x^3 - 6*x^4 + 16*x^5 - 31*x^6 + 126*x^7 - 492*x^8 + 1180*x^9 - 4092*x^10 + 17972*x^11 - 52498*x^12 + ...

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

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

%e (2) R(x*(1+x)^4*A(x)) = x^2*(1 + x)^5.

%e GENERATING METHOD.

%e Define F(n), a polynomial in x of order 7^(n-1), by the following recurrence:

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

%e F(2) = (1 + x^2 * (1+x)^5),

%e F(3) = (1 + x^4 * (1+x)^10 * F(2)^5),

%e F(4) = (1 + x^8 * (1+x)^20 * F(2)^10 * F(3)^5),

%e F(5) = (1 + x^16 * (1+x)^40 * F(2)^20 * F(3)^10 * F(4)^5),

%e ...

%e F(n+1) = 1 + (F(n) - 1)^2 * F(n)^5

%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)^5 ) - x*(1 + x)^4*F ,#A+1) ); A[n]}

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

%Y Cf. A369555, A369545, A369546, A369547, A369549.

%K nonn

%O 1,4

%A _Paul D. Hanna_, Jan 25 2024