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A360949 G.f. A(x) satisfies: 1 = Sum_{n>=0} (-x/2)^n * (A(x)^n + (-1)^n)^n. 2

%I #15 Mar 05 2023 12:37:19

%S 1,2,8,50,376,3124,27804,260496,2539616,25556330,263922884,2785341186,

%T 29948035032,327315887046,3630399545244,40813503158790,

%U 464662514679168,5354222585965310,62419468527625408,736098528973804246,8781173950238637928,105987886325647341056

%N G.f. A(x) satisfies: 1 = Sum_{n>=0} (-x/2)^n * (A(x)^n + (-1)^n)^n.

%C Conjecture: a(0) = 1, a(2*A161674(k) + 1) == 2 (mod 4) for k >= 1, otherwise a(n) == 0 (mod 4). A161674 lists positions n such that A010060(n) + A010060(n+2) = 1, where A010060 is the Thue-Morse sequence.

%H Paul D. Hanna, <a href="/A360949/b360949.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

%F (1) 1 = Sum_{n>=0} (-x/2)^n * (A(x)^n + (-1)^n)^n.

%F (2) 1 = Sum_{n>=0} 2 * (-x)^n * A(x)^(n^2) / (2 - (-1)^n * x * A(x)^n)^(n+1).

%F a(n) = A325574(n)/2^n for n >= 0.

%e G.f.: A(x) = 1 + 2*x + 8*x^2 + 50*x^3 + 376*x^4 + 3124*x^5 + 27804*x^6 + 260496*x^7 + 2539616*x^8 + 25556330*x^9 + ...

%e such that

%e 1 = 1 - (x/2)*(A(x) - 1) + (x/2)^2*(A(x)^2 + 1)^2 - (x/2)^3*(A(x)^3 - 1)^3 + (x/2)^4*(A(x)^4 + 1)^4 - (x/2)^5*(A(x)^5 - 1)^5 + (x/2)^6*(A(x)^6 + 1)^6 - (x/2)^7*(A(x)^7 - 1)^7 + ...

%e also,

%e 1 = 2/(2 - x) - 2*x*A(x)/(2 + x*A(x))^2 + 2*x^2*A(x)^4/(2 - x*A(x)^2)^3 - 2*x^3*A(x)^9/(2 + x*A(x)^3)^4 + 2*x^4*A(x)^16/(2 - x*A(x)^4)^5 - 2*x^5*A(x)^25/(2 + x*A(x)^5)^6 + 2*x^6*A(x)^36/(2 - x*A(x)^6)^7 ... + 2*(-x)^n*A(x)^(n^2)/(2 - (-1)^n*x*A(x)^n)^(n+1) + ...

%o (PARI) {a(n) = my(A=[1]);

%o for(i=1, n, A=concat(A, 0); A[#A] = 2*polcoeff( sum(m=0, #A, (-x/2)^m * (Ser(A)^m + (-1)^m)^m ), #A)); A[n+1]}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A325574, A161674.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 03 2023

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