A363558 Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (2 + x^n)^(2*n).

1, 5, 16, 77, 256, 1104, 4121, 16832, 65536, 264688, 1048617, 4205568, 16779008, 67162112, 268436016, 1073999165, 4294967296, 17180983296, 68719549696, 274882887680, 1099511628896, 4398068904021, 17592186086656, 70368840646656, 281474978676736, 1125900326286464

Offset: 0

Author

Paul D. Hanna, Aug 01 2023

Keywords

• nonn

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds as a formal power series for all y.

Examples

			G.f.: A(x) = 1 + 5*x + 16*x^2 + 77*x^3 + 256*x^4 + 1104*x^5 + 4121*x^6 + 16832*x^7 + 65536*x^8 + 264688*x^9 + 1048617*x^10 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A260147, A363559, A363569, A363561.

Programs

PARI

{a(n) = my(A); A = sum(m=-n-1,n+1, x^m * (2 + x^m +x*O(x^n))^(2*m) ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

The g.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.

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

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

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

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

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

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

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

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

From Paul D. Hanna, Aug 06 2023: (Start)

The following generating functions are extensions of Peter Bala's formulas given in A260147.

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

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

(End)

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

a(p) = p*2^(p-1) + 4^p for primes p > 3 (conjecture).