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A176335 Central coefficients T(2n,n) of number triangle A176331.

%I A176335 #28 Nov 29 2024 19:55:04
%S A176335 1,3,28,315,3876,50358,678112,9365499,131809060,1882294128,
%T A176335 27193657008,396600597198,5829739893264,86262567856650,
%U A176335 1283677784658528,19196304797150715,288295493121264420,4346056823245242420
%N A176335 Central coefficients T(2n,n) of number triangle A176331.
%H A176335 G. C. Greubel, <a href="/A176335/b176335.txt">Table of n, a(n) for n = 0..825</a>
%H A176335 F. Baldassarri, S. Bosch, B. Dwork, (eds), <a href="https://doi.org/10.1007/BFb0091139">p-adic Analysis. Lecture Notes in Mathematics</a>, vol. 1454, pp. 194 - 204,  Springer, Berlin, Heidelberg.
%H A176335 Matthijs J. Coster, <a href="https://core.ac.uk/download/pdf/301642554.pdf">Supercongruences</a>.
%F A176335 a(n) = Sum_{k=0..2n} C(k,n)^2*(-1)^k.
%F A176335 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
%F A176335 From _Peter Bala_, Aug 08 2024: (Start)
%F A176335 a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n+k, k)^2. Cf. A112029.
%F A176335 Conjecture (assuming an offset of 1): the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) holds for all primes p >= 5 and all positive integers n and r [added Nov 29 2024: proved by Coster. See Theorem 4]. (End)
%F A176335 a(n) ~ 2^(4*n+2) / (5*Pi*n). - _Vaclav Kotesovec_, Aug 08 2024
%F A176335 a(n) = binomial(2*n, n)^2 * hypergeom([1, -n, -n], [-2*n, -2*n], -1). - _Peter Luschny_, Nov 29 2024
%p A176335 A176335 := proc(n)
%p A176335     add((-1)^k*binomial(k,n)^2,k=0..2*n);
%p A176335 end proc: # _R. J. Mathar_, Feb 10 2015
%t A176335 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 A176335 (PARI) T(n,k) = sum(j=0, n, (-1)^(n-j)*binomial(j, n-k)*binomial(j, k));
%o A176335 vector(31, n, T(2*(n-1), n-1) ) \\ _G. C. Greubel_, Dec 07 2019
%o A176335 (Magma) T:= func< n,k | &+[(-1)^(n-j)*Binomial(j,n-k)*Binomial(j,k): j in [0..n]] >;
%o A176335 [T(2*n,n): n in [0..30]]; // _G. C. Greubel_, Dec 07 2019
%o A176335 (Sage)
%o A176335 @CachedFunction
%o A176335 def T(n, k): return sum( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k) for j in (0..n))
%o A176335 [T(2*n, n) for n in (0..30)] # _G. C. Greubel_, Dec 07 2019
%o A176335 (GAP)
%o A176335 T:= function(n,k)
%o A176335     return Sum([0..n], j-> (-1)^(n-j)*Binomial(j,k)*Binomial(j,n-k) );
%o A176335   end;
%o A176335 List([0..30], n-> T(2*n,n) ); # _G. C. Greubel_, Dec 07 2019
%Y A176335 Cf. A112029, A176331, A176332, A176334.
%K A176335 nonn,easy
%O A176335 0,2
%A A176335 _Paul Barry_, Apr 15 2010