A272199 Expansion of 1/(1 - 2*x + 13*x^2).

1, 2, -9, -44, 29, 630, 883, -6424, -24327, 34858, 385967, 318780, -4380011, -12904162, 31131819, 230017744, 55321841, -2879586990, -6478357913, 24477915044, 133174482957, -51863929658, -1834996137757, -2995761189960, 17863427410921, 74671750291322

Offset: 0

Author

Wolfdieter Lang, Apr 27 2016

Keywords

• sign
• easy

Comments

a(n) gives the coefficient c(13^n) of (eta(z^6))^4, a modular cusp form of weight 2, when expanded in powers of q = exp(2*Pi*i*z), Im(z) > 0, assuming alpha-multiplicativity (not valid for p = 2 and 3) with alpha(x) = x (weight 2) and input c(13) = +2. Eta is the Dedekind function. See the Apostol reference, p. 138, eq. (54) for alpha-multiplicativity and p. 130, eq. (39) with k=2. See also A000727 where a(n)=c(13^n) = A000727((13^n-1)/6)=A000727(2*A091030(n)), n >= 0. For the proof that alpha-multiplicativity leads to the recurrence involving Chebyshev's S polynomials see a comment on A168175, and the Apostol reference, Exercise 6., p. 139.

Examples

			a(2) = c(13^2) = A000727(2*A091030(2)) =
A000727(28) = -9.
		

References

• Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, pp. 130, 138 - 139.

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..1790
• Index entries for linear recurrences with constant coefficients, signature (2,-13).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A023000, A049310, A168175.

Programs

Magma

I:=[1,2]; [n le 2 select I[n] else 2*Self(n-1)-13*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 25 2016

Mathematica

CoefficientList[Series[1/(1 - 2 x + 13 x^2), {x, 0, 25}], x] (* Michael De Vlieger, Apr 27 2016 *)
LinearRecurrence[{2, -13}, {1, 2}, 30] (* Vincenzo Librandi, Nov 25 2016 *)

PARI

Vec(1/(1-2*x+13*x^2) + O(x^99)) \\ Altug Alkan, Apr 28 2016

Formula

G.f.: 1/(1 - 2*x + 13*x^2).

a(n) = 2*a(n-1) - 13*a(n-2), a(-1) = 0, a(0) = 1.

a(n) = sqrt(13)^n * S(n, 2/sqrt(13)), n >= 0, with Chebyshev's S polynomials (A049310).

a(n) = (Ap^(n+1) - Am^(n+1))/(Ap - Am) with Ap:= 1 + 2*sqrt(3)*i and Am = 1 - 2*sqrt(3)*i, (Binet - de Moivre formula), where i is the imaginary unit.

E.g.f.: (sqrt(3)*sin(2*sqrt(3)*x) + 6*cos(2*sqrt(3)*x))*exp(x)/6. - Ilya Gutkovskiy, Apr 27 2016