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A275757 G.f. satisfies: A(x) = x + A( A(x)^3 - A(x)^7 ), an odd function.

%I A275757 #21 Feb 15 2019 09:58:21
%S A275757 1,1,3,11,46,207,977,4767,23835,121424,627747,3284055,17348254,
%T A275757 92387544,495371637,2671588333,14480158111,78822638280,430685654483,
%U A275757 2361012092488,12980509646385,71547277918984,395252428706918,2187886348193235,12132382884810469,67383306100049693,374771558921409855,2086989709106321626,11634599273439782284,64923785744439199536,362598744217074249165,2026617482659866472677
%N A275757 G.f. satisfies: A(x) = x + A( A(x)^3 - A(x)^7 ), an odd function.
%C A275757 Compare to: F(x) = x + F( F(x)^3 - F(x)^9 ) holds when F(x) = x + F(x)^3 is a g.f. of the ternary tree numbers (A001764).
%C A275757 Compare to: C(x) = x + C( C(x)^2 - C(x)^4 ) holds when C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
%C A275757 First negative term is a(83). _Georg Fischer_, Feb 15 2019
%H A275757 Paul D. Hanna, <a href="/A275757/b275757.txt">Table of n, a(n) for n = 1..200</a>
%F A275757 G.f. satisfies:
%F A275757 (1) A(x - A(x^3 - x^7)) = x.
%F A275757 (2) A(x) = x + Sum_{n>=0} d^n/dx^n A(x^3-x^7)^(n+1) / (n+1)!.
%F A275757 (3) A(x) = x * exp( Sum_{n>=0} d^n/dx^n A(x^3-x^7)^(n+1)/x / (n+1)! ).
%e A275757 G.f.: A(x) = x + x^3 + 3*x^5 + 11*x^7 + 46*x^9 + 207*x^11 + 977*x^13 + 4767*x^15 + 23835*x^17 + 121424*x^19 + 627747*x^21 + 3284055*x^23 + 17348254*x^25 +...
%e A275757 such that A(x) = x + A( A(x)^3 - A(x)^7 ).
%e A275757 RELATED SERIES.
%e A275757 A(x)^3 = x^3 + 3*x^5 + 12*x^7 + 52*x^9 + 240*x^11 + 1155*x^13 + 5727*x^15 + 29034*x^17 + 149727*x^19 + 782627*x^21 + 4135668*x^23 + 22051158*x^25 +...
%e A275757 A(x)^7 = x^7 + 7*x^9 + 42*x^11 + 238*x^13 + 1323*x^15 + 7308*x^17 + 40327*x^19 + 222804*x^21 + 1233624*x^23 + 6847281*x^25 + 38102099*x^27 +...
%e A275757 A(x^3 - x^7) = x^3 - x^7 + x^9 - 3*x^13 + 3*x^15 + 3*x^17 - 15*x^19 + 10*x^21 + 30*x^23 - 77*x^25 + 16*x^27 + 231*x^29 - 399*x^31 - 178*x^33 + 1653*x^35 - 1892*x^37 - 2887*x^39 +...
%e A275757 where Series_Reversion(A(x)) = x - A(x^3 - x^7).
%o A275757 (PARI) {a(n) = my(A=x); for(i=1, 2*n, A = x + subst(A, x, A^3 - A^7 +x*O(x^(2*n)))); polcoeff(A, 2*n-1)}
%o A275757 for(n=1, 30, print1(a(n), ", "))
%Y A275757 Cf. A275755, A275756, A275758, A179486.
%K A275757 sign
%O A275757 1,3
%A A275757 _Paul D. Hanna_, Aug 20 2016