A363569 Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (i + x^n)^(2*n), where i^2 = -1.

1, 0, 1, -3, 1, 4, -4, -8, 1, 23, -8, -12, -27, 12, 36, 15, 1, 16, -149, -20, 71, 91, 166, -24, -119, -186, -285, 251, -26, 28, 761, -32, 1, -199, -679, -827, -310, 36, 970, 1572, 1821, 40, -2631, -44, 331, -5628, 1772, -48, -495, 3051, 2546, 6697, -1715, 52, 2791, -7425, -8007, -10869, -3653, -60

Offset

0,4

Comments

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

Links

Paul D. Hanna, Table of n, a(n) for n = 0..4100

Formula

The g.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas; here, i = (+/-) sqrt(-1).

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

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

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

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

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

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

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

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

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

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

(End)

a(2^n) = 1 for n > 0 (conjecture).

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

Example

G.f.: A(x) = 1 + x^2 - 3*x^3 + x^4 + 4*x^5 - 4*x^6 - 8*x^7 + x^8 + 23*x^9 - 8*x^10 - 12*x^11 - 27*x^12 + 12*x^13 + 36*x^14 + 15*x^15 + ...

Prog

(PARI) {a(n) = my(A); A = sum(m=-n-1, n+1, x^m * (I + x^m +x*O(x^n))^(2*m) ); polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))

Crossrefs

Cf. A260147, A363558, A363559.

Sequence in context: A021322 A277129 A036412 * A200027 A370807 A298890

Adjacent sequences: A363566 A363567 A363568 * A363570 A363571 A363572

Keyword

sign

Author

Paul D. Hanna, Jul 31 2023

Status

approved