A363314 Expansion of g.f. A(x) satisfying 1/4 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1), with a(0) = 5.

5, 32, 496, 9024, 181296, 3882848, 86887712, 2007577472, 47530180736, 1147071160768, 28114384217104, 697913487791552, 17511114852998912, 443374443981736160, 11314170816869911232, 290688529521060711424, 7513202655833624201472, 195216134898681278515232

Offset: 0

Author

Paul D. Hanna, May 28 2023

Keywords

• nonn

Comments

a(n) == 0 (mod 4^2) for n > 0.

Examples

			G.f.: A(x) =  5 + 32*x + 496*x^2 + 9024*x^3 + 181296*x^4 + 3882848*x^5 + 86887712*x^6 + 2007577472*x^7 + 47530180736*x^8 + ...
		

Links

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

Crossrefs

Cf. A357227, A363141, A363312, A363313, A363315.

Programs

PARI

{a(n) = my(A=[5]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(-4  + 4^2*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) 1/4 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1).

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

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

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