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A305135
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G.f. A(x) satisfies: 1 = Sum_{n>=0} (x^n - A(x))^n.
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2
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1, 1, -1, -4, -2, 18, 40, -30, -289, -346, 1151, 4319, 658, -27009, -55352, 76422, 521723, 474100, -2701432, -8516598, 2522919, 65164743, 108748950, -254553384, -1279831072, -665253375, 7987959534, 20547223364, -18007631653, -186391127742, -235366676097, 912066105170, 3550265215708, 259991506989, -25935784814796, -53475930511840, 87779402689817, 575950379415007, 488006663591490
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OFFSET
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1,4
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} (x^n - A(x))^n.
(2) 1 = Sum_{n>=0} x^(n^2) / (1 + x^n*A(x))^(n+1).
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EXAMPLE
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G.f.: A(x) = x + x^2 - x^3 - 4*x^4 - 2*x^5 + 18*x^6 + 40*x^7 - 30*x^8 - 289*x^9 - 346*x^10 + 1151*x^11 + 4319*x^12 + 658*x^13 - 27009*x^14 - 55352*x^15 + 76422*x^16 + 521723*x^17 + 474100*x^18 - 2701432*x^19 - 8516598*x^20 + 2522919*x^21 + 65164743*x^22 + 108748950*x^23 - 254553384*x^24 - 1279831072*x^25 + ...
such that
1 = 1 + (x - A(x)) + (x^2 - A(x))^2 + (x^3 - A(x))^3 + (x^4 - A(x))^4 + (x^5 - A(x))^5 + (x^6 - A(x))^6 + (x^7 - A(x))^7 + ... + (x^n - A(x))^n + ...
Also,
1 = 1/(1 + A(x)) + x/(1 + x*A(x))^2 + x^4/(1 + x^2*A(x))^3 + x^9/(1 + x^3*A(x))^4 + x^16/(1 + x^4*A(x))^5 + x^25/(1 + x^5*A(x))^6 + x^36/(1 + x^6*A(x))^7 + x^49/(1 + x^7*A(x))^8 + x^64/(1 + x^8*A(x))^9 + ... + x^(n^2)/(1 + x^n*A(x))^(n+1) + ...
RELATED SERIES.
Series_Reversion(A(x)) = x - x^2 + 3*x^3 - 6*x^4 + 16*x^5 - 46*x^6 + 142*x^7 - 471*x^8 + 1606*x^9 - 5616*x^10 + 19946*x^11 + ... + A305136(n)*x^n + ...
SPECIFIC VALUES.
(1) Although the series diverges at x = 1/2, it can be evaluated formally.
Let b = A(1/2) = 0.545218973635949431234950245034944106957612798888179456724264...
then 1 = Sum_{n>=0} (1/2^n - b)^n
and 1 = Sum_{n>=0} 2^n/(2^n + b)^(n+1).
(2) The series converges at x = 1/3 ;
let t = A(1/3) = 0.3773536554501605670840638296726444190565890261333229000698630...
then 1 = Sum_{n>=0} (1/3^n - t)^n
and 1 = Sum_{n>=0} 3^n/(3^n + t)^(n+1).
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PROG
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(PARI) /* From 1 = Sum_{n>=0} (x^n - A(x))^n */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A+1, (x^m - x*Ser(A))^m ) )[#A+1] ); A[n]}
for(n=1, 30, print1(a(n), ", "))
(PARI) /* From 1 = Sum_{n>=0} x^(n^2) / (1 + x^n * A(x))^(n+1) */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, sqrtint(#A+1), x^(m^2)/(1 + x^m*x*Ser(A))^(m+1) ) )[#A+1] ); A[n]}
for(n=1, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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