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%I #9 Aug 15 2023 18:52:31
%S 1,1,3,16,99,670,4804,35855,275635,2167577,17354844,140994899,
%T 1159398760,9631155422,80703507043,681333999628,5789823864323,
%U 49484286592503,425092050147999,3668385302806058,31786451503719132,276447315011186576,2412336247105063011,21114946136742383146
%N Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.
%F (1) A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).
%F (2) A(x)^2 = 1 + x*(A(x) + A(x)^2 - A(x)^3 - A(x)^4 + A(x)^5 + A(x)^6).
%F (3) A(x)^3 = 1 + x*(A(x) + A(x)^2 - A(x)^4 + A(x)^6 + A(x)^7).
%F (4) A(x)^4 = 1 + x*(A(x) + A(x)^2 + A(x)^7 + A(x)^8).
%F (5) A(x)^5 = 1 + x*(A(x) + A(x)^2 + A(x)^5 + A(x)^8 + A(x)^9).
%F (6) A(x)^6 = 1 + x*(A(x) + A(x)^2 + A(x)^5 + A(x)^6 + A(x)^9 + A(x)^10).
%F (7) A(x)^7 = 1 + x*(A(x) + A(x)^2 + A(x)^5 + A(x)^6 + A(x)^7 + A(x)^10 + A(x)^11).
%F (8) A(x) = (1/x) * Series_Reversion( x/(1 + Series_Reversion( x/(1 + x*(1+x)^2 + x*(1+x)^3) ) ) ).
%F (9) A(x) = 1 / A(-x*A(x)^5).
%e G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 99*x^4 + 670*x^5 + 4804*x^6 + 35855*x^7 + 275635*x^8 + 2167577*x^9 + 17354844*x^10 + ...
%e such that
%e A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).
%e RELATED TABLE.
%e The table of coefficients in A(x)^n begins:
%e n=1: [1, 1, 3, 16, 99, 670, 4804, 35855, 275635, ...];
%e n=2: [1, 2, 7, 38, 239, 1634, 11798, 88506, 683045, ...];
%e n=3: [1, 3, 12, 67, 429, 2967, 21594, 162945, 1263183, ...];
%e n=4: [1, 4, 18, 104, 679, 4756, 34922, 265244, 2066591, ...];
%e n=5: [1, 5, 25, 150, 1000, 7101, 52645, 402725, 3155125, ...];
%e n=6: [1, 6, 33, 206, 1404, 10116, 75775, 584148, 4603911, ...];
%e n=7: [1, 7, 42, 273, 1904, 13930, 105490, 819918, 6503553, ...];
%e n=8: [1, 8, 52, 352, 2514, 18688, 143152, 1122312, 8962615, ...];
%e n=9: [1, 9, 63, 444, 3249, 24552, 190326, 1505727, 12110400, ...];
%e ...
%e from which one can verify the formulas involving powers of A(x).
%e RELATED SERIES.
%e Let G(x) = 1 + Series_Reversion( x/(1 + x*(1+x)^2 + x*(1+x)^3) )
%e where
%e G(x) = 1 + x + 2*x^2 + 9*x^3 + 42*x^4 + 219*x^5 + 1202*x^6 + 6867*x^7 + 40378*x^8 + 242782*x^9 + 1485836*x^10 + ...
%e then
%e A(x) = G(x*A(x)),
%e and so
%e A(x) = (1/x) * Series_Reversion( x/G(x) );
%e thus,
%e x*A(x) = (A(x) - 1) / (1 + (A(x) - 1)*(A(x)^2 + A(x)^3) )
%e which is equivalent to
%e A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).
%p a:= n-> coeff(series(RootOf(1-A+x*(A-A^3+A^5), A), x, n+1), x, n):
%p seq(a(n), n=0..23); # _Alois P. Heinz_, Aug 14 2023
%o (PARI) {a(n) = my(A=1+x); for(i=1, n, A = 1 + x*(A - A^3 + A^5) +x*O(x^n) ); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A219537, A271469, A300048, A161634, A363560.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Aug 14 2023
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