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A181147 a(n) = Sum_{k=0..n-1} (105k+44)*C(2k,k)^2*T(k)*(-1)^(n-1-k)/(2n*C(2n,n)), where T(k) (k=0,1,2,...) are central trinomial coefficients given by A002426.

%I A181147 #22 Nov 04 2019 02:25:24
%S A181147 11,23,224,1747,16754,162392,1651206,17126327,181182446,1943132842,
%T A181147 21080299228,230802972664,2546569337336,28280754214358,
%U A181147 315824396838386,3544003431783795,39936833763112790,451718158386620678
%N A181147 a(n) = Sum_{k=0..n-1} (105k+44)*C(2k,k)^2*T(k)*(-1)^(n-1-k)/(2n*C(2n,n)), where T(k) (k=0,1,2,...) are central trinomial coefficients given by A002426.
%C A181147 On Jan 22 2011, _Zhi-Wei Sun_ conjectured that a(n) is a positive integer for every n=1,2,3,... Let p > 3 be a prime. He also conjectured that a(p) == 5 + 6*(p/3)*(2-3^(p-1)) (mod p^2). Another conjecture of his states that Sum_{k=0..p-1} (-1)^k*binomial(2k,k)^2*T(k) is congruent to b(p) modulo p^2, where b(p)=0 if (p/15)=-1, b(p) = 4x^2-2p if p == 1,4 (mod 15) and p = x^2+15y^2 with x,y integers, and b(p) = 20x^2-2p if p == 2,8 (mod 15) and p=5x^2+3y^2 with x,y integers.
%H A181147 D. S. McNeil, <a href="/A181147/b181147.txt">Table of n, a(n) for n = 1..900</a>
%H A181147 Zhi-Wei Sun, <a href="http://arxiv.org/abs/0911.5665">Open Conjectures on Congruences</a>, preprint, arXiv:0911.5665 [math.NT], 2009-2011.
%H A181147 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1101.0600">On sums related to central binomial and trinomial coefficients</a>, preprint, arXiv:1101.0600 [math.NT], 2011-2014.
%e A181147 For n=2 we have a(2) = (44*1^2*T(0)(-1) + (105+44)*2^2*T(1))/(2*2*binomial(4,2)) = 23.
%p A181147 T:= n-> `if`(n=0, 1, coeff ((x^2+x+1)^n, x, n)):
%p A181147 a:= n-> add ((105*k+44) *binomial (2*k, k)^2 *T(k)*(-1)^(n-1-k),
%p A181147              k=0..n-1)/ (2*n*binomial (2*n,  n)):
%p A181147 seq (a(n), n=1..30);
%t A181147 T[k_]:=If[k>0,Coefficient[(x^2+x+1)^k,x^k],1]
%t A181147 A[n_]:=Sum[(105k+44)Binomial[2k,k]^2*T[k](-1)^(n-1-k),{k,0,n-1}]/(2n*Binomial[2n,n])
%t A181147 Table[A[n],{n,1,50}]
%Y A181147 Cf. A000984, A002426, A173774.
%K A181147 nonn
%O A181147 1,1
%A A181147 _Zhi-Wei Sun_, Jan 24 2011