A144133 Gegenbauer polynomial C_n^2(3).

1, 12, 106, 828, 6051, 42408, 288788, 1925736, 12637733, 81897876, 525360702, 3341936196, 21109664455, 132544828560, 827948567080, 5148653356944, 31891223012553, 196848686563164, 1211273655997202, 7432579805359884

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Eric Weisstein's World of Mathematics, Gegenbauer Polynomial.

Wikipedia, Gegenbauer polynomials

Index entries for linear recurrences with constant coefficients, signature (12,-38,12,-1)

Formula

From Michael Somos, May 11 2012: (Start)

G.f.: 1 / (1 - 6*x + x^2)^2.

a(-4 - n) = -a(n).

Convolution square of A001109. (End)

From Emanuele Munarini, Mar 07 2018: (Start)

a(n) = (1/4)*Sum_{k=0..n} p(2*k+1)*p(2*n-2*k+1) = (1/32)*(14*n+13)*p(2*n+1) + (3/16)*(n+1)*p(2*n), where the p(n) = A000129(n+1) are Pell numbers.

a(n+4) - 12*a(n+3) + 38*a(n+2) - 12*a(n+1) + a(n) = 0. (End)

Example

1 + 12*x + 106*x^2 + 828*x^3 + 6051*x^4 + 42408*x^5 + ...

Mathematica

lst={}; Do[AppendTo[lst, GegenbauerC[n, 2, 3]], {n, 0, 8^3}]; lst
LinearRecurrence[{12, -38, 12, -1}, {1, 12, 106, 828}, 100] (* Emanuele Munarini, Mar 07 2018 *)

Prog

(PARI) {a(n) = local(s=1); if( n<0, n = -4 - n; s=-1); s * polcoeff( 1 / (1 - 6*x + x^2)^2 + x * O(x^n), n)} /* Michael Somos, May 11 2012 */
(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -1, 12, -38, 12]^n*[1; 12; 106; 828])[1, 1] \\ Charles R Greathouse IV, Feb 07 2022
(Maxima) makelist(ultraspherical(n, 2, 3), n, 0, 24); /* Emanuele Munarini, Mar 07 2018 */

Crossrefs

Cf. A000129, A001109.

Sequence in context: A027142 A090816 A244722 * A089396 A218111 A166755

Adjacent sequences: A144130 A144131 A144132 * A144134 A144135 A144136

Keyword

nonn,easy

Author

Vladimir Joseph Stephan Orlovsky, Sep 11 2008

Status

approved