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Gegenbauer polynomial C_n^2(3).
2

%I #32 Feb 16 2025 08:33:09

%S 1,12,106,828,6051,42408,288788,1925736,12637733,81897876,525360702,

%T 3341936196,21109664455,132544828560,827948567080,5148653356944,

%U 31891223012553,196848686563164,1211273655997202,7432579805359884

%N Gegenbauer polynomial C_n^2(3).

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GegenbauerPolynomial.html">Gegenbauer Polynomial</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Gegenbauer_polynomials">Gegenbauer polynomials</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (12,-38,12,-1)

%F From _Michael Somos_, May 11 2012: (Start)

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

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

%F Convolution square of A001109. (End)

%F From _Emanuele Munarini_, Mar 07 2018: (Start)

%F 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.

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

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

%t lst={};Do[AppendTo[lst,GegenbauerC[n,2,3]],{n,0,8^3}];lst

%t LinearRecurrence[{12, -38, 12, -1}, {1, 12, 106, 828}, 100] (* _Emanuele Munarini_, Mar 07 2018 *)

%o (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 */

%o (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

%o (Maxima) makelist(ultraspherical(n,2,3),n,0,24); /* _Emanuele Munarini_, Mar 07 2018 */

%Y Cf. A000129, A001109.

%K nonn,easy,changed

%O 0,2

%A _Vladimir Joseph Stephan Orlovsky_, Sep 11 2008