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

1, 3, 28, 315, 3876, 50358, 678112, 9365499, 131809060, 1882294128, 27193657008, 396600597198, 5829739893264, 86262567856650, 1283677784658528, 19196304797150715, 288295493121264420, 4346056823245242420

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..825

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

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

Cf. A176331, A176332, A176334.

Sequence in context: A250890 A199754 A277507 * A072343 A292845 A212032

Adjacent sequences: A176332 A176333 A176334 * A176336 A176337 A176338

Keyword

easy,nonn

Author

Paul Barry, Apr 15 2010

Status

approved