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%I #14 Sep 08 2022 08:45:52
%S 1,3,28,315,3876,50358,678112,9365499,131809060,1882294128,
%T 27193657008,396600597198,5829739893264,86262567856650,
%U 1283677784658528,19196304797150715,288295493121264420,4346056823245242420
%N Central coefficients T(2n,n) of number triangle A176331.
%H G. C. Greubel, <a href="/A176335/b176335.txt">Table of n, a(n) for n = 0..825</a>
%F a(n) = Sum_{k=0..2n} C(k,n)^2*(-1)^k.
%F Conjecture: 224*n^2*(n-1)*a(n) - 48*(n-1)*(65*n^2 - 36*n - 13)*a(n-1) + 4*(-1839*n^3 + 11081*n^2 - 21932*n + 14280)*a(n-2) + 12*(-81*n^3 + 326*n^2 - 591*n + 562)*a(n-3) - (n-3)*(1853*n^2 - 7403*n + 7140)*a(n-4) - 12*(n-4)*(2*n-7)^2*a(n-5) = 0. - _R. J. Mathar_, Feb 10 2015
%p A176335 := proc(n)
%p add((-1)^k*binomial(k,n)^2,k=0..2*n);
%p end proc: # _R. J. Mathar_, Feb 10 2015
%t T[n_, k_]:= Sum[(-1)^(n-j)*Binomial[j, k]*Binomial[j, n-k], {j,0,n}]; Table[T[2*n, n], {n,0,30}] (* _G. C. Greubel_, Dec 07 2019 *)
%o (PARI) T(n,k) = sum(j=0, n, (-1)^(n-j)*binomial(j, n-k)*binomial(j, k));
%o vector(31, n, T(2*(n-1), n-1) ) \\ _G. C. Greubel_, Dec 07 2019
%o (Magma) T:= func< n,k | &+[(-1)^(n-j)*Binomial(j,n-k)*Binomial(j,k): j in [0..n]] >;
%o [T(2*n,n): n in [0..30]]; // _G. C. Greubel_, Dec 07 2019
%o (Sage)
%o @CachedFunction
%o def T(n, k): return sum( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k) for j in (0..n))
%o [T(2*n, n) for n in (0..30)] # _G. C. Greubel_, Dec 07 2019
%o (GAP)
%o T:= function(n,k)
%o return Sum([0..n], j-> (-1)^(n-j)*Binomial(j,k)*Binomial(j,n-k) );
%o end;
%o List([0..30], n-> T(2*n,n) ); # _G. C. Greubel_, Dec 07 2019
%Y Cf. A176331, A176332, A176334.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Apr 15 2010
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