A025170 Expansion of g.f.: 1/(1 + 2*x + 9*x^2).

1, -2, -5, 28, -11, -230, 559, 952, -6935, 5302, 51811, -151340, -163619, 1689298, -1906025, -11391632, 39937489, 22649710, -404736821, 605626252, 2431378885, -10313394038, -1255621889, 95331790120, -179362983239, -499260144602, 2612787138355, -732232975292

Offset: 0

Author

Wouter Meeussen

Keywords

• sign
• easy

Comments

Reciprocal Chebyshev polynomial of second kind evaluated at 3 multiplied by (-1)^n.

From Sharon Sela (sharonsela(AT)hotmail.com), Jan 19 2002: (Start)

a(n) is (-1)^n times the determinant of the following tridiagonal n X n matrix:

[2 3 0 . . . . . . .]

[3 2 3 0 . . . . . .]

[0 3 2 3 0 . . . . .]

[. 0 3 2 3 0 . . . .]

[. . . . . . . . . .]

[. . . . . . . . . .]

[. . . . 0 3 2 3 0 .]

[. . . . . 0 3 2 3 0]

[. . . . . . 0 3 2 3]

[. . . . . . . 0 3 2]

(End)

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-2,-9).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Variant is A127357.

Cf. A087455, A088137.

Programs

Magma

[(-3)^n*Evaluate(ChebyshevU(n+1),1/3): n in [0..50]]; // G. C. Greubel, Jan 02 2024

Mathematica

Table[3^n ChebyshevU[n, -1/3], {n, 0, 24}]

PARI

a(n)=if(n<0,0,polcoeff(1/(1+2*x+9*x^2)+x*O(x^n),n))

PARI

a(n)=if(n<0, 0, 3^n*subst(poltchebi(n+1)+3*poltchebi(n),'x,-1/3)*3/8) /* Michael Somos, Sep 15 2005 */

PARI

a(n)=if(n<0, 0, (-1)^n*matdet(matrix(n,n,i,j, if(abs(i-j)<2, 2+abs(i-j))))) /* Michael Somos, Sep 15 2005 */

SageMath

[3^n*chebyshev_U(n,-1/3) for n in range(41)] # G. C. Greubel, Jan 02 2024

Formula

a(n) = 3^n * ChebyshevU(n, -1/3).

a(n) = ( A088137(n+1) )^2 + ( A087455(n+1)/2 )^2 - ( A087455(n+2)/2 )^2. - Creighton Dement, Aug 20 2004

a(n) = -(2*a(n-1) + 9*a(n-2)) for n>1, with a(0)=1, a(1)=-2. - Philippe Deléham, Sep 19 2009

a(n) = (-2)^n*Product_{k=1..n}(1 + 3*cos(k*Pi/(n+1))). - Peter Luschny, Nov 28 2019

From G. C. Greubel, Jan 02 2024: (Start)

a(n) = (-1)^n * A127357(n).

E.g.f.: (1/4)*exp(-x)*(4*cos(2*sqrt(2)*x) - sqrt(2)*sin(2*sqrt(2)*x)). (End)