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 A237645 G.f. satisfies: A(x) = G(x*A(x)) where G(x) = -1+x + A(x) + 1/A(x). 1
 1, 1, 2, 7, 34, 201, 1357, 10109, 81397, 698948, 6341597, 60391832, 600661215, 6215862360, 66726103981, 741259084280, 8504902411004, 100618874020119, 1225724374602147, 15356200178917791, 197646961110310062, 2610956607315266757, 35370366025297098315 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..200 FORMULA G.f. satisfies: (1) A(x) = -1 + x*A(x) + A(x*A(x)) + 1/A(x*A(x)). (2) A(x) = (1/x) * Series_Reversion( x / ( -1+x + A(x) + 1/A(x) ) ). a(n) = [x^n] ( -1+x + A(x) + 1/A(x) )^(n+1) / (n+1). EXAMPLE G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 34*x^4 + 201*x^5 + 1357*x^6 +... Let G(x) = -1+x + A(x) + 1/A(x): G(x) = 1 + x + x^2 + 3*x^3 + 13*x^4 + 70*x^5 + 436*x^6 + 3024*x^7 + 22828*x^8 + 184795*x^9 + 1587809*x^10 +... then A(x) = G(x*A(x)) and G(x) = A(x/G(x)). Related expansions. A(x*A(x)) = 1 + x + 3*x^2 + 13*x^3 + 72*x^4 + 470*x^5 + 3449*x^6 + 27662*x^7 + 238209*x^8 + 2176591*x^9 + 20928935*x^10 +... 1/A(x*A(x)) = 1 - x - 2*x^2 - 8*x^3 - 45*x^4 - 303*x^5 - 2293*x^6 - 18910*x^7 - 166921*x^8 - 1559040*x^9 - 15286286*x^10 +... where A(x) = -1 + x*A(x) + A(x*A(x)) + 1/A(x*A(x)). PROG (PARI) {a(n)=local(A=[1, 1]); for(m=2, n+1, A[m]=Vec((-1+x+ Ser(A) +1/Ser(A))^m)[m]/m; A=concat(A, 0)); A[n+1]} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Sequence in context: A199475 A241599 A307696 * A117399 A145345 A212027 Adjacent sequences:  A237642 A237643 A237644 * A237646 A237647 A237648 KEYWORD nonn AUTHOR Paul D. Hanna, May 02 2014 STATUS approved

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Last modified January 29 05:09 EST 2022. Contains 350672 sequences. (Running on oeis4.)