A363312 Expansion of g.f. A(x) satisfying 1/2 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1), with a(0) = 3.

3, 8, 68, 656, 6924, 77816, 912504, 11043616, 136909712, 1729812880, 22193496988, 288368706416, 3786876943856, 50180784019384, 670150485880336, 9010466250798080, 121871951481594296, 1657086342551799752, 22637216782139196588, 310547100988853539728

Offset: 0

Author

Paul D. Hanna, May 28 2023

Keywords

• nonn

Comments

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

Examples

			G.f.: A(x) = 3 + 8*x + 68*x^2 + 656*x^3 + 6924*x^4 + 77816*x^5 + 912504*x^6 + 11043616*x^7 + 136909712*x^8 + 1729812880*x^9 + ...
		

Links

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

Crossrefs

Cf. A357227, A363141, A363313, A363314, A363315.

Programs

PARI

{a(n) = my(A=[3]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(-2  + 2^2*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) 1/2 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1).

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

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

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