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A176335
Central coefficients T(2n,n) of number triangle A176331.
4
1, 3, 28, 315, 3876, 50358, 678112, 9365499, 131809060, 1882294128, 27193657008, 396600597198, 5829739893264, 86262567856650, 1283677784658528, 19196304797150715, 288295493121264420, 4346056823245242420
OFFSET
0,2
LINKS
FORMULA
a(n) = Sum_{k=0..2n} C(k,n)^2*(-1)^k.
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
From Peter Bala, Aug 08 2024: (Start)
a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n+k, k)^2. Cf. A112029.
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. (End)
a(n) ~ 2^(4*n+2) / (5*Pi*n). - Vaclav Kotesovec, Aug 08 2024
MAPLE
A176335 := proc(n)
add((-1)^k*binomial(k, n)^2, k=0..2*n);
end proc: # R. J. Mathar, Feb 10 2015
MATHEMATICA
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 *)
PROG
(PARI) T(n, k) = sum(j=0, n, (-1)^(n-j)*binomial(j, n-k)*binomial(j, k));
vector(31, n, T(2*(n-1), n-1) ) \\ G. C. Greubel, Dec 07 2019
(Magma) T:= func< n, k | &+[(-1)^(n-j)*Binomial(j, n-k)*Binomial(j, k): j in [0..n]] >;
[T(2*n, n): n in [0..30]]; // G. C. Greubel, Dec 07 2019
(Sage)
@CachedFunction
def T(n, k): return sum( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k) for j in (0..n))
[T(2*n, n) for n in (0..30)] # G. C. Greubel, Dec 07 2019
(GAP)
T:= function(n, k)
return Sum([0..n], j-> (-1)^(n-j)*Binomial(j, k)*Binomial(j, n-k) );
end;
List([0..30], n-> T(2*n, n) ); # G. C. Greubel, Dec 07 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Apr 15 2010
STATUS
approved