A326265 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1/(1-x)^(5*n) - A(x) )^n.

1, 5, 40, 1185, 65270, 4861126, 445776670, 48124064710, 5952881626790, 828544320379330, 128058593506875627, 21758230559633783765, 4031357498037096661170, 809070343591564791211705, 174888309616496370413590235, 40517215307075701804767255261, 10017278630199891781122121185615, 2632883558256463087445119555912870, 733167697272377998186394054589647855, 215641985221691590110546294934099963285

Offset: 0

Author

Paul D. Hanna, Jun 20 2019

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 5*x + 40*x^2 + 1185*x^3 + 65270*x^4 + 4861126*x^5 + 445776670*x^6 + 48124064710*x^7 + 5952881626790*x^8 + 828544320379330*x^9 + 128058593506875627*x^10 + ...
such that
1 = 1  +  (1/(1-x)^5 - A(x))  +  (1/(1-x)^10 - A(x))^2  +  (1/(1-x)^15 - A(x))^3  +  (1/(1-x)^20 - A(x))^4  +  (1/(1-x)^25 - A(x))^5  +  (1/(1-x)^30 - A(x))^6  +  (1/(1-x)^35 - A(x))^7  + ...
Also,
1 = 1/(1 + A(x))  +  (1-x)^5/((1-x)^5 + A(x))^2  +  (1-x)^10/((1-x)^10 + A(x))^3  +  (1-x)^15/((1-x)^15  +  A(x))^4 + (1-x)^20/((1-x)^20 + A(x))^5  +  (1-x)^25/((1-x)^25 + A(x))^6  +  (1-x)^30/((1-x)^30 + A(x))^7 + ...
		

Crossrefs

Cf. A304639, A326262, A326263, A326264.

Cf. A321605.

Programs

PARI

{a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1-x)^(-5*m) - Ser(A))^m ) )[#A] ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} ( 1/(1-x)^(5*n) - A(x) )^n.

(2) 1 = Sum_{n>=0} ( 1 - (1-x)^(5*n)*A(x) )^n / (1-x)^(5*n^2).

(3) 1 = Sum_{n>=0} (1-x)^(5*n) / ( (1-x)^(5*n) + A(x) )^(n+1).