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A098470 Form array in which n-th row is obtained by expanding (1+x+x^2)^n and taking the 5th column from the center.

%I A098470 #18 Feb 16 2025 08:32:54
%S A098470 1,6,28,112,414,1452,4917,16236,52624,168168,531531,1665456,5182008,
%T A098470 16031952,49366674,151419816,462919401,1411306358,4292487562,
%U A098470 13029127584,39478598170,119439969220,360881425710,1089126806040
%N A098470 Form array in which n-th row is obtained by expanding (1+x+x^2)^n and taking the 5th column from the center.
%H A098470 G. C. Greubel, <a href="/A098470/b098470.txt">Table of n, a(n) for n = 5..1005</a>
%H A098470 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient</a>
%F A098470 (n^2-25)*a(n) = n*(2*n-1)*a(n-1) + 3*n*(n-1)*a(n-2). - _Vladeta Jovovic_, Sep 18 2004
%F A098470 G.f.: 32*x^5/(sqrt((1+x)*(1-3*x))*(1-x-sqrt((1+x)*(1-3*x)))^5). - _Vladeta Jovovic_, Sep 18 2004
%F A098470 a(n) = A111808(n,n-5). - _Reinhard Zumkeller_, Aug 17 2005
%F A098470 Assuming offset 0: a(n) = GegenbauerC(n,-n-5,-1/2) and a(n) = binomial(10+2*n,n)* hypergeom([-n, -n-10], [-9/2-n], 1/4). - _Peter Luschny_, May 09 2016
%F A098470 a(n) ~ 3^(n + 1/2) / (2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Nov 09 2021
%p A098470 # Assuming offset 0:
%p A098470 a := n -> simplify(GegenbauerC(n, -n-5, -1/2)):
%p A098470 seq(a(n), n=0..25); # _Peter Luschny_, May 09 2016
%t A098470 Table[GegenbauerC[n, -n - 5, -1/2], {n,0,50}] (* _G. C. Greubel_, Feb 28 2017 *)
%o A098470 (PARI) x='x + O('x^50); Vec(32*x^5/(sqrt((1+x)*(1-3*x))*(1-x-sqrt((1+x)*(1-3*x)))^5)) \\ _G. C. Greubel_, Feb 28 2017
%Y A098470 Cf. A002426, A005717, A014531, A014532, A014533.
%K A098470 nonn
%O A098470 5,2
%A A098470 _Eric W. Weisstein_, Sep 09 2004