A355865 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * (x^n - (-1)^n*2*A(x))^(2*n+1).

1, 3, 25, 254, 2844, 34031, 426498, 5526399, 73433377, 995167783, 13701794657, 191122323160, 2695092314319, 38357425655599, 550268824751092, 7948720164361366, 115517358604881329, 1687796954715824015, 24777722054035138573, 365305177280838473896

Offset: 0

Author

Paul D. Hanna, Aug 04 2022

Keywords

• nonn

Comments

Compare to the identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n which holds for all y.

Examples

			G.f.: A(x) = 1 + 3*x + 25*x^2 + 254*x^3 + 2844*x^4 + 34031*x^5 + 426498*x^6 + 5526399*x^7 + 73433377*x^8 + 995167783*x^9 + 13701794657*x^10 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A355866.

Programs

PARI

{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, x^m * (x^m - (-1)^m*2*Ser(A))^(2*m+1) ), #A-1)/2);A[n+1]}
for(n=0,20,print1(a(n),", "))

Formula

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

(1) 0 = Sum_{n=-oo..+oo} (-x)^n * ((-x)^n - 2*A(x))^(2*n+1).

(2) 0 = Sum_{n=-oo..+oo} x^(2*n*(n-1)) / (1 - 2*A(x)*(-x)^n)^(2*n-1).