A363561 G.f.: Sum_{n=-oo..+oo} x^n * (sqrt(2) + x^n)^(2*n).

1, 3, 4, 15, 16, 52, 77, 184, 256, 716, 1045, 2400, 4320, 9024, 16524, 35439, 65536, 135424, 264928, 534016, 1048856, 2124523, 4196944, 8435712, 16792576, 33658512, 67118016, 134478584, 268435513, 537346048, 1073876144, 2148499456, 4294967296, 8592337520, 17179956224, 34364358760

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 + 3*x + 4*x^2 + 15*x^3 + 16*x^4 + 52*x^5 + 77*x^6 + 184*x^7 + 256*x^8 + 716*x^9 + 1045*x^10 + 2400*x^11 + 4320*x^12 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1030

Crossrefs

Cf. A260147, A363558, A363559, A363569.

Programs

PARI

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

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

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

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

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

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

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

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

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

(End)

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

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