A363559 Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (3 + x^n)^(2*n).

1, 10, 81, 757, 6561, 59454, 531496, 4788072, 43046721, 387480753, 3486784492, 31381709148, 282429556893, 2541872737062, 22876792457796, 205891204134565, 1853020188851841, 16677182431460826, 150094635300957591, 1350851725033981380, 12157665459056934471

Offset

0,2

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..300

Formula

The g.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

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

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

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

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

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

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

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

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

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

(End)

a(2^n) = 9^(2^n) for n > 0 (conjecture).

a(p) = p*3^(p-1) + 9^p for primes p > 3 (conjecture).

Example

G.f.: A(x) = 1 + 10*x + 81*x^2 + 757*x^3 + 6561*x^4 + 59454*x^5 + 531496*x^6 + 4788072*x^7 + 43046721*x^8 + 387480753*x^9 + ...

Prog

(PARI) {a(n) = my(A); A = sum(m=-n-1, n+1, x^m * (3 + x^m +x*O(x^n))^(2*m) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A260147, A363558, A363569, A363561.

Sequence in context: A037541 A037485 A343276 * A350503 A277205 A308862

Adjacent sequences: A363556 A363557 A363558 * A363560 A363561 A363562

Keyword

nonn

Author

Paul D. Hanna, Aug 01 2023

Status

approved