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%I #20 Aug 28 2017 07:35:25
%S 1,2,2,6,40,208,798,3122,15038,77830,381798,1819998,8925172,45280900,
%T 231030138,1171823534,5962836408,30668699312,158951012362,
%U 825830001086,4298605879552,22459588992656,117842770179898,620193719988230,3271151667546526,17291851589803030,91629268113394082,486633483668452306,2589396122840425628,13802082307489152876,73692343820697785462,394098991084750746722
%N G.f. A(x) satisfies: A( 2*A(x)^2 - 4*A(x)^3 ) = 2*x^2 + 4*x^3.
%C Series reversion of the g.f. is described by A290958.
%H Paul D. Hanna, <a href="/A290957/b290957.txt">Table of n, a(n) for n = 1..520</a>
%F a(n) ~ c * d^n / n^(3/2), where d = 5.6107024438745823... and c = 0.07583382126393587... - _Vaclav Kotesovec_, Aug 28 2017
%e G.f.: A(x) = x + 2*x^2 + 2*x^3 + 6*x^4 + 40*x^5 + 208*x^6 + 798*x^7 + 3122*x^8 + 15038*x^9 + 77830*x^10 + 381798*x^11 + 1819998*x^12 + 8925172*x^13 + 45280900*x^14 + 231030138*x^15 + 1171823534*x^16 + 5962836408*x^17 + 30668699312*x^18 + 158951012362*x^19 + 825830001086*x^20 +...
%e such that A( 2*A(x)^2 - 4*A(x)^3 ) = 2*x^2 + 4*x^3.
%e Let B(x) be the series reversion of A(x), then B(x) is the g.f. of A290958 and begins;
%e B(x) = x - 2*x^2 + 6*x^3 - 26*x^4 + 100*x^5 - 460*x^6 + 2258*x^7 - 11558*x^8 + 60786*x^9 - 326826*x^10 + 1785930*x^11 - 9893778*x^12 + 55447800*x^13 - 313817720*x^14 + 1791442406*x^15 - 10303155322*x^16 +...+ A290958(n)*x^n +...
%e where B( 2*x^2 + 4*x^3 ) = 2*A(x)^2 - 4*A(x)^3,
%e also, A( 2*x^2 - 4*x^3 ) = 2*B(x)^2 + 4*B(x)^3,
%e and B( 2*B(x)^2 + 4*B(x)^3 ) = 2*x^2 - 4*x^3.
%e Related series begin:
%e 2*A(x)^2 - 4*A(x)^3 = 2*x^2 - 12*x^3 + 56*x^4 - 272*x^5 + 1312*x^6 - 6432*x^7 + 32640*x^8 - 170576*x^9 + 911696*x^10 - 4963760*x^11 + 27425200*x^12 +...
%e 2*B(x)^2 + 4*B(x)^3 = 2*x^2 + 12*x^3 + 40*x^4 + 112*x^5 + 416*x^6 + 2112*x^7 + 10336*x^8 + 45936*x^9 + 206192*x^10 + 999376*x^11 + 5026640*x^12 +...
%o (PARI) /* Informal code to generate N terms */
%o {C=[1,2]; for(i=1,N=60,
%o A = sum(n=1,#C,C[n]*x^(n) ) + t*x^(#C+1) +O(x^(#C+2));
%o S = subst(A,x, 2*A^2 - 4*A^3);
%o C = concat(C,polcoeff(subst(-S/deriv(polcoeff(S,#C+2,x),t),t,0),#C+2,x) )); C}
%Y Cf. A290958 (inverse), A271961.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Aug 14 2017
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