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A179089 a(n) = (1/n^2) * Sum_{k=0..n-1} (2k+1)*T_k^2(-3)^(n-1-k), where T_0, T_1, ... are central trinomial coefficients given by A002426.

%I A179089 #29 Nov 13 2022 14:31:05
%S A179089 1,0,5,13,105,576,4005,27000,193193,1402672,10433709,78807785,
%T A179089 603996745,4683970032,36702939429,290184446349,2312460578025,
%U A179089 18556825469040,149842592021997,1216719520281045,9929612901775761,81406058258856240
%N A179089 a(n) = (1/n^2) * Sum_{k=0..n-1} (2k+1)*T_k^2(-3)^(n-1-k), where T_0, T_1, ... are central trinomial coefficients given by A002426.
%C A179089 On Jun 28 2010, _Zhi-Wei Sun_ conjectured that a(n) is an integer for every n=1,2,3,... and that a(p) = (1+p/3)/2 (mod p) for any prime p, where (p/3) is the Legendre symbol. In contrast, he showed that Sum_{k=0..n-1} (2k+1)*T_k^2*3^(n-1-k) = n*T_n*T_{n-1} for all n=1,2,3,...
%C A179089 The formula for a(n) in the formula section implies that a(n) is an integer.  - _Mark van Hoeij_, Nov 13 2022
%H A179089 G. C. Greubel, <a href="/A179089/b179089.txt">Table of n, a(n) for n = 1..1000</a>
%H A179089 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1006.2776">Arithmetic properties of Apery numbers and central Delannoy numbers</a>, preprint, arXiv:1006.2776 [math.NT], 2010-2011.
%F A179089 G.f.: Integral(hypergeom([1/2, 3/2], [2], 16*x/(1 + 3*x)^2)/(1 + 3*x)^2). - _Mark van Hoeij_, Nov 10 2022
%F A179089 a(n) = (A002426(n)+A002426(n-1))*(3*A002426(n-1)-A002426(n))/4. - _Mark van Hoeij_, Nov 13 2022
%e A179089 For n = 4 we have a(4) = (T_0^2(-3)^3 + 3*T_1^2(-3)^2 + 5*T_2^2(-3) + 7*T_3^2)/4^2 = (-27 + 27 - 5*27 + 7^3)/16 = 13.
%p A179089 A002426 := n -> simplify(GegenbauerC(n, -n, -1/2)); seq( (A002426(n)+A002426(n-1))*(3*A002426(n-1)-A002426(n))/4, n=1..20); # _Mark van Hoeij_, Nov 13 2022
%t A179089 TT[n_]:=Sum[Binomial[n,2k]Binomial[2k,k],{k,0,Floor[n/2]}] SS[n_]:=Sum[(2k+1)*TT[k]^2*(-3)^(n-1-k),{k,0,n-1}]/n^2 Table[SS[n],{n,1,50}]
%Y A179089 Cf. A002426, A178808, A178790, A178791, A173774.
%K A179089 nonn
%O A179089 1,3
%A A179089 _Zhi-Wei Sun_, Jun 29 2010