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 A307999 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n * (1 - (-1)^n*A(x))^(n+1), where A(0) = 0. 0

%I #6 May 09 2019 16:19:59

%S 1,3,5,15,65,255,961,3759,15233,62655,260097,1090623,4616769,19698687,

%T 84611841,365570559,1587755777,6928284927,30358910977,133532161023,

%U 589348292609,2609230704639,11584885657601,51571340750847,230129898799105,1029215591587839,4612514610282497,20711143725961215,93164104646180865,419778524769746943,1894404146662522881,8561776644482695167

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

%F G.f. A = A(x) satisfies

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

%F (2) A = (1 + A)^2*x + ((1 + A)^3 - 4*A)*x^2 - (1 - A^2)^2*x^3 - (1 - A^2)^2*x^4.

%F (3) 0 = x^3*(1+x)*A^4 - x^2*A^3 - (2*x^4 + 2*x^3 + 3*x^2 + x)*A^2 + (1-x)^2*A - x*(1-x)*(1+x)^2.

%e G.f.: A(x) = x + 3*x^2 + 5*x^3 + 15*x^4 + 65*x^5 + 255*x^6 + 961*x^7 + 3759*x^8 + 15233*x^9 + 62655*x^10 + 260097*x^11 + 1090623*x^12 + ...

%e such that

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

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

%o for(n=1,40,print1(a(n),", "))

%K nonn

%O 1,2

%A _Paul D. Hanna_, May 09 2019

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Last modified May 22 02:24 EDT 2024. Contains 372741 sequences. (Running on oeis4.)