A361778 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * ((-x)^(n-1) - 2*A(x))^n.
1, 2, 7, 27, 109, 459, 2006, 9017, 41384, 193048, 912571, 4361939, 21045710, 102361864, 501349447, 2470556294, 12240270901, 60935582862, 304660949343, 1529125824203, 7701783889261, 38915600049447, 197206343307012, 1002023916642621, 5103911800972155, 26056404563941575
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 2*x + 7*x^2 + 27*x^3 + 109*x^4 + 459*x^5 + 2006*x^6 + 9017*x^7 + 41384*x^8 + 193048*x^9 + 912571*x^10 + ... SPECIFIC VALUES. A(1/7) = 1.63053651133635034184414884744745628155427916612173429157... A(1/6) = 1.99892384479086071017436459041327119822244448085100733509... A(x) = 2 at x = 0.166713109990638926829644490786806133084979604287174064... Radius of convergence r and the value A(r) are given by r = 0.182033752413024354859591633469061831146023401652842514076551... A(r) = 2.63999965897091399750291467200041973752650665197493948118984006... 1/r = 5.4934867119096473651972990947886642212447897087082048838...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..300
Programs
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PARI
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=-#A, #A, x^m * ((-x)^(m-1) - 2*Ser(A))^m ), #A)/2); A[n+1]} for(n=0, 30, print1(a(n), ", "))
Formula
G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) 1 = Sum_{n=-oo..+oo} x^n * ((-x)^(n-1) - 2*A(x))^n.
(2) 1 = Sum_{n=-oo..+oo} x^(2*n+1) * (2*A(x) - (-x)^n)^n.
(3) 2*A(x) = Sum_{n=-oo..+oo} x^(3*n+1) * ((-x)^n - 2*A(x))^n.
(4) 1 = Sum_{n=-oo..+oo} x^(n^2) / (1 - 2*A(x)*(-x)^(n+1))^n.
(5) 1 = Sum_{n=-oo..+oo} x^(n^2) / (1 - 2*A(x)*(-x)^(n+1))^(n+1).
(6) 2*A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - 2*A(x)*(-x)^(n+1))^(n+1).
(7) 0 = Sum_{n=-oo..+oo} x^(2*n) * (2*A(x) - (-x)^n)^(n+1).
(8) 0 = Sum_{n=-oo..+oo} x^(3*n) * ((-x)^(n-1) - 2*A(x))^(n+1).