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A176335 Central coefficients T(2n,n) of number triangle A176331. 3
1, 3, 28, 315, 3876, 50358, 678112, 9365499, 131809060, 1882294128, 27193657008, 396600597198, 5829739893264, 86262567856650, 1283677784658528, 19196304797150715, 288295493121264420, 4346056823245242420 (list; graph; refs; listen; history; text; internal format)
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

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Last modified June 12 18:42 EDT 2021. Contains 344959 sequences. (Running on oeis4.)