A363315 Expansion of g.f. A(x) satisfying 1/5 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1), with a(0) = 6.

6, 50, 950, 21350, 530700, 14067650, 389701050, 11147799700, 326779719500, 9764739197800, 296342706620800, 9108989853295550, 283002934668287000, 8872796279035164100, 280368062326854982450, 8919740526808334086550, 285476263708658548421000, 9185078302539674382641450

Offset: 0

Author

Paul D. Hanna, May 28 2023

Keywords

• nonn

Comments

a(n) == 0 (mod 5^2) for n > 0.

Examples

			G.f.: A(x) = 6 + 50*x + 950*x^2 + 21350*x^3 + 530700*x^4 + 14067650*x^5 + 389701050*x^6 + 11147799700*x^7 + 326779719500*x^8 + ...
		

Links

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

Crossrefs

Cf. A357227, A363141, A363312, A363313, A363314.

Programs

PARI

{a(n) = my(A=[6]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(-5  + 5^2*sum(m=-#A, #A, x^m * (Ser(A) - x^m)^(m-1) ), #A-1););A[n+1]}
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.

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

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

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

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