A379205 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 5)^(n+1).

1, 7, 74, 998, 15268, 251427, 4345869, 77751128, 1427455842, 26740178711, 509068777424, 9820550568868, 191554931918517, 3771529984556599, 74857068226445132, 1496158969938529383, 30086862802675119068, 608303992207446069349, 12358069554479794052292, 252144178158939689795128

Offset: 1

Author

Paul D. Hanna, Dec 20 2024

Keywords

• nonn

Examples

			G.f.: A(x) = x + 7*x^2 + 74*x^3 + 998*x^4 + 15268*x^5 + 251427*x^6 + 4345869*x^7 + 77751128*x^8 + 1427455842*x^9 + 26740178711*x^10 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..230

Crossrefs

Cf. A379200, A379199, A166952, A378264, A379202, A379203, A379204.

Programs

PARI

{a(n) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A, #A, A^m*(A^m + 5)^(m+1) ), #V-3); ); polcoef(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

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

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

(3) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + 5*A(x)^(n+1))^n.

(4) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 - 5*A(x)^(n+1))^(n+1).

(5) A(B(x)) = x where B(x) = 1/( Sum_{n=-oo..+oo} x^n * (x^n + 5)^(n+1) ).