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 A304639 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1/(1-x)^n - A(x) )^n. 4
 1, 1, 2, 11, 117, 1735, 31853, 689043, 17079221, 476238926, 14742680162, 501584454703, 18605089712174, 747393133162471, 32332767332220442, 1498961537925543920, 74153115616699819304, 3899494667155151052688, 217246028175467702590241, 12783023090792392539557926, 792236994094236725330142276, 51585659784100723438219893047, 3520987513029712770759434038820 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..200 EXAMPLE G.f. A(x) satisfies: (1) 1 = Sum_{n>=0} ( 1/(1-x)^n - A(x) )^n. (2) 1 = Sum_{n>=0} ( 1 - (1-x)^n*A(x) )^n / (1-x)^(n^2). (3) 1 = Sum_{n>=0} (1-x)^n / ( (1-x)^n + A(x) )^(n+1). MAPLE G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 117*x^4 + 1735*x^5 + 31853*x^6 + 689043*x^7 + 17079221*x^8 + 476238926*x^9 + 14742680162*x^10 + 501584454703*x^11 + ... such that 1 = 1  +  (1/(1-x) - A(x))  +  (1/(1-x)^2 - A(x))^2  +  (1/(1-x)^3 - A(x))^3  +  (1/(1-x)^4 - A(x))^4  +  (1/(1-x)^5 - A(x))^5  +  (1/(1-x)^6 - A(x))^6  +  (1/(1-x)^7 - A(x))^7  + ... Also, 1 = 1/(1 + A(x))  +  (1-x)/((1-x) + A(x))^2  +  (1-x)^2/((1-x)^2 + A(x))^3  +  (1-x)^3/((1-x)^3  +  A(x))^4 + (1-x)^4/((1-x)^4 + A(x))^5  +  (1-x)^5/((1-x)^5 + A(x))^6  +  (1-x)^6/((1-x)^6 + A(x))^7 + ... PARTICULAR VALUES. Although the power series A(x) diverges at x = -1, it may be evaluated formally. Let t = A(-1) = 0.5452189736359494312349502450349441069576127988881794567242641... then t satisfies (1) 1 = Sum_{n>=0} ( 1/2^n - t )^n. (2) 1 = Sum_{n>=0} ( 1 - 2^n*t )^n / 2^(n^2). (3) 1 = Sum_{n>=0} 2^n / ( 2^n + t )^(n+1). PROG (PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, (1/(1-x +x^2*O(x^n))^m - Ser(A))^m ) )[#A] ); A[n+1]} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A303056. Sequence in context: A286869 A181168 A269082 * A130222 A197993 A057076 Adjacent sequences:  A304636 A304637 A304638 * A304640 A304641 A304642 KEYWORD nonn AUTHOR Paul D. Hanna, May 16 2018 STATUS approved

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Last modified March 23 16:52 EDT 2019. Contains 321432 sequences. (Running on oeis4.)