A363573 - OEIS

%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