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A176013 Triangle, read by rows, T(n, k) = (-1)^n * n!/(k*k!) * binomial(n-1, k-1) * binomial(n, k-1). 5

%I

%S -1,2,1,-6,-9,-1,24,72,24,1,-120,-600,-400,-50,-1,720,5400,6000,1500,

%T 90,1,-5040,-52920,-88200,-36750,-4410,-147,-1,40320,564480,1317120,

%U 823200,164640,10976,224,1,-362880,-6531840,-20321280,-17781120,-5334336,-592704,-24192,-324,-1

%N Triangle, read by rows, T(n, k) = (-1)^n * n!/(k*k!) * binomial(n-1, k-1) * binomial(n, k-1).

%C Row sums are: -1, 3, -16, 121, -1171, 13711, -187468, 2920961, -50948677, 981458011, ...

%H G. C. Greubel, <a href="/A176013/b176013.txt">Rows n = 1..100 of the triangle, flattened</a>

%F T(n, k) = (-1)^n * n!/(k*k!) * binomial(n-1, k-1) * binomial(n, k-1).

%F T(n, k) = binomial(n+1, k) * A008297(n, k)/(n+1). - _G. C. Greubel_, Feb 08 2021

%e Triangle begins as:

%e -1;

%e 2, 1;

%e -6, -9, -1;

%e 24, 72, 24, 1;

%e -120, -600, -400, -50, -1;

%e 720, 5400, 6000, 1500, 90, 1;

%e -5040, -52920, -88200, -36750, -4410, -147, -1;

%e 40320, 564480, 1317120, 823200, 164640, 10976, 224, 1;

%e -362880, -6531840, -20321280, -17781120, -5334336, -592704, -24192, -324, -1;

%t T[n_, k_] = (-1)^n*n!/(k*k!)*Binomial[n-1, k-1]*Binomial[n, k-1];

%t Table[T[n, k], {n,12}, {k,n}]//Flatten

%o (Sage) flatten([[(-1)^n*(factorial(n)/(k*factorial(k)))*binomial(n-1, k-1)*binomial(n, k-1) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Feb 08 2021

%o (Magma) [(-1)^n*(Factorial(n)/(k*Factorial(k)))*Binomial(n-1, k-1)*Binomial(n, k-1) : k in [1..n], n in [1..12]]; // _G. C. Greubel_, Feb 08 2021

%Y Cf. A008297.

%K sign,tabl,easy,less

%O 1,2

%A _Roger L. Bagula_, Apr 06 2010

%E Edited by _G. C. Greubel_, Feb 08 2021

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Last modified October 20 10:43 EDT 2021. Contains 348100 sequences. (Running on oeis4.)