proposed

approved

editing

proposed

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

Row sums are -1, 1, 4, 12, 32, 80, 192, 448, 1024, 2304, 5120,.. see A001787.

Sum_{k=0..n} T(n,k) = A001787(n), n >= 1.

(Maxima) create_list(n*binomial(n - 1, k) - (-1)^(n - k)*binomial(n, k), n , 0, 15, k, 0, n); /* Franck Maminirina Ramaharo, Jan 25 2019 */

Cf. A001787, A007318, A130595, A144388, A216973.

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

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

Triangle begins:

-1;

2, -1;

1, 4, -1;

4, 3, 6, -1;

3, 16, 6, 8, -1;

6, 15, 40, 10, 10, -1;

5, 36, 45, 80, 15, 12, -1;

8, 35, 126, 105, 140, 21, 14, -1;

7, 64, 140, 336, 210, 224, 28, 16, -1;

10, 63, 288, 420, 756, 378, 336, 36, 18, -1;

9, 100, 315, 960, 1050, 1512, 630, 480, 45, 20, -1;

...

Clear[p, x, n]; p[x_, n_] = -(x - 1)^n + n*(x + 1)^(n - 1); Table[ExpandAll[p[x, n]], {n, 0, 10}]; Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[%]

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

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

sign,easy,tabl

approved

editing

proposed

approved

editing

proposed

A triangle of coefficients for polynomials: p(x,n)=-(x - 1)^n + n*(x + 1)^(n - 1).

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

1,0,2

Row sums are: -1, 1, 4, 12, 32, 80, 192, 448, 1024, 2304, 5120,.. see A001787.

{-1, 1, 4, 12, 32, 80, 192, 448, 1024, 2304, 5120}.

p(x,n)=-(x - 1)^n + n*(x + 1)^(n - 1); t(n,m)=coefficients(p(x,n)).

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

{-1},

-1;

{2, -1},;

{1, 4, -1},;

{4, 3, 6, -1},;

{3, 16, 6, 8, -1},;

{6, 15, 40, 10, 10, -1},;

{5, 36, 45, 80, 15, 12, -1},;

{8, 35, 126, 105, 140, 21, 14, -1},;

{7, 64, 140, 336, 210, 224, 28, 16, -1},;

{10, 63, 288, 420, 756, 378, 336, 36, 18, -1},;

{9, 100, 315, 960, 1050, 1512, 630, 480, 45, 20, -1};

sign,unedtabl

approved

editing

_Roger L. Bagula _ and _Gary W. Adamson (rlbagulatftn(AT)yahoo.com), _, Oct 01 2008

A triangle of coefficients for polynomials: p(x,n)=-(x - 1)^n + n*(x + 1)^(n - 1).

-1, 2, -1, 1, 4, -1, 4, 3, 6, -1, 3, 16, 6, 8, -1, 6, 15, 40, 10, 10, -1, 5, 36, 45, 80, 15, 12, -1, 8, 35, 126, 105, 140, 21, 14, -1, 7, 64, 140, 336, 210, 224, 28, 16, -1, 10, 63, 288, 420, 756, 378, 336, 36, 18, -1, 9, 100, 315, 960, 1050, 1512, 630, 480, 45, 20, -1

1,2

Row sums are:

{-1, 1, 4, 12, 32, 80, 192, 448, 1024, 2304, 5120}.

p(x,n)=-(x - 1)^n + n*(x + 1)^(n - 1); t(n,m)=coefficients(p(x,n)).

{-1},

{2, -1},

{1, 4, -1},

{4, 3, 6, -1},

{3, 16, 6, 8, -1},

{6, 15, 40, 10, 10, -1},

{5, 36, 45, 80, 15, 12, -1},

{8, 35, 126, 105, 140, 21, 14, -1},

{7, 64, 140, 336, 210, 224, 28, 16, -1},

{10, 63, 288, 420, 756, 378, 336, 36, 18, -1},

{9, 100, 315, 960, 1050, 1512, 630, 480, 45, 20, -1}

Clear[p, x, n]; p[x_, n_] = -(x - 1)^n + n*(x + 1)^(n - 1); Table[ExpandAll[p[x, n]], {n, 0, 10}]; Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[%]

sign,uned

Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Oct 01 2008

approved