%I #7 Jan 26 2019 04:59:55

%S 1,0,1,3,0,1,-2,9,0,1,5,-8,18,0,1,-4,25,-20,30,0,1,7,-24,75,-40,45,0,

%T 1,-6,49,-84,175,-70,63,0,1,9,-48,196,-224,350,-112,84,0,1,-8,81,-216,

%U 588,-504,630,-168,108,0,1,11,-80,405,-720,1470,-1008,1050,-240,135,0,1

%N Triangle T(n,k) = binomial(n, k) + ((-1)^(n + k))*n*binomial(n - 1, k), T(0,0) = 1, read by rows, 0 <= k <= n.

%F T(n,k) = [x^k] ((x + 1)^n - n*(x - 1)^(n - 1)).

%F Sum_{k=0..n} T(n,k) = A151821(n-1), n >= 1.

%e Triangle begins:

%e 1;

%e 0, 1;

%e 3, 0, 1;

%e -2, 9, 0, 1;

%e 5, -8, 18, 0, 1;

%e -4, 25, -20, 30, 0, 1;

%e 7, -24, 75, -40, 45, 0, 1;

%e -6, 49, -84, 175, -70, 63, 0, 1;

%e 9, -48, 196, -224, 350, -112, 84, 0, 1;

%e -8, 81, -216, 588, -504, 630, -168, 108, 0, 1;

%e 11, -80, 405, -720, 1470, -1008, 1050, -240, 135, 0, 1;

%e ...

%t p[x_, n_] = (x + 1)^n - n*(x - 1)^(n - 1);

%t Table[CoefficientList[p[x, n], x], {n, 0, 10}] // Flatten

%o (Maxima) create_list(binomial(n, k) + ((-1)^(n + k))*n*binomial(n - 1, k),n , 0, 15, k, 0, n); /* _Franck Maminirina Ramaharo_, Jan 25 2019 */

%Y Cf. A001787, A151821, A144389, A216973.

%K sign,easy,tabl

%O 0,4

%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 01 2008

%E Edited and offset corrected by _Franck Maminirina Ramaharo_, Jan 25 2019