A363573 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).

1, 1, 3, 16, 99, 670, 4804, 35855, 275635, 2167577, 17354844, 140994899, 1159398760, 9631155422, 80703507043, 681333999628, 5789823864323, 49484286592503, 425092050147999, 3668385302806058, 31786451503719132, 276447315011186576, 2412336247105063011, 21114946136742383146

Offset: 0

Author

Paul D. Hanna, Aug 14 2023

Keywords

• nonn

Examples

			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 + ...
such that
A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).
RELATED TABLE.
The table of coefficients in A(x)^n begins:
n=1: [1, 1,  3,  16,   99,   670,   4804,   35855,   275635, ...];
n=2: [1, 2,  7,  38,  239,  1634,  11798,   88506,   683045, ...];
n=3: [1, 3, 12,  67,  429,  2967,  21594,  162945,  1263183, ...];
n=4: [1, 4, 18, 104,  679,  4756,  34922,  265244,  2066591, ...];
n=5: [1, 5, 25, 150, 1000,  7101,  52645,  402725,  3155125, ...];
n=6: [1, 6, 33, 206, 1404, 10116,  75775,  584148,  4603911, ...];
n=7: [1, 7, 42, 273, 1904, 13930, 105490,  819918,  6503553, ...];
n=8: [1, 8, 52, 352, 2514, 18688, 143152, 1122312,  8962615, ...];
n=9: [1, 9, 63, 444, 3249, 24552, 190326, 1505727, 12110400, ...];
...
from which one can verify the formulas involving powers of A(x).
RELATED SERIES.
Let G(x) = 1 + Series_Reversion( x/(1 + x*(1+x)^2 + x*(1+x)^3) )
where
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 + ...
then
A(x) = G(x*A(x)),
and so
A(x) = (1/x) * Series_Reversion( x/G(x) );
thus,
x*A(x) = (A(x) - 1) / (1 + (A(x) - 1)*(A(x)^2 + A(x)^3) )
which is equivalent to
A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).
		

Crossrefs

Cf. A219537, A271469, A300048, A161634, A363560.

Programs

Maple

a:= n-> coeff(series(RootOf(1-A+x*(A-A^3+A^5), A), x, n+1), x, n):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 14 2023

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)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.

(1) A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).

(2) A(x)^2 = 1 + x*(A(x) + A(x)^2 - A(x)^3 - A(x)^4 + A(x)^5 + A(x)^6).

(3) A(x)^3 = 1 + x*(A(x) + A(x)^2 - A(x)^4 + A(x)^6 + A(x)^7).

(4) A(x)^4 = 1 + x*(A(x) + A(x)^2 + A(x)^7 + A(x)^8).

(5) A(x)^5 = 1 + x*(A(x) + A(x)^2 + A(x)^5 + A(x)^8 + A(x)^9).

(6) A(x)^6 = 1 + x*(A(x) + A(x)^2 + A(x)^5 + A(x)^6 + A(x)^9 + A(x)^10).

(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).

(8) A(x) = (1/x) * Series_Reversion( x/(1 + Series_Reversion( x/(1 + x*(1+x)^2 + x*(1+x)^3) ) ) ).

(9) A(x) = 1 / A(-x*A(x)^5).