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 A305135 G.f. A(x) satisfies: 1 = Sum_{n>=0} (x^n - A(x))^n. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 LINKS Paul D. Hanna, Table of n, a(n) for n = 1..1030 FORMULA 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). EXAMPLE 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). PROG (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), ", ")) CROSSREFS Cf. A305136. Sequence in context: A285793 A117692 A052966 * A177248 A191706 A081797 Adjacent sequences: A305132 A305133 A305134 * A305136 A305137 A305138 KEYWORD sign AUTHOR Paul D. Hanna, May 30 2018 STATUS approved

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Last modified June 2 04:18 EDT 2023. Contains 363081 sequences. (Running on oeis4.)