A141692 Triangle read by rows: T(n,k) = n*(binomial(n - 1, k - 1) - binomial(n - 1, k)), 0 <= k <= n.

0, -1, 1, -2, 0, 2, -3, -3, 3, 3, -4, -8, 0, 8, 4, -5, -15, -10, 10, 15, 5, -6, -24, -30, 0, 30, 24, 6, -7, -35, -63, -35, 35, 63, 35, 7, -8, -48, -112, -112, 0, 112, 112, 48, 8, -9, -63, -180, -252, -126, 126, 252, 180, 63, 9, -10, -80, -270, -480, -420, 0, 420, 480, 270, 80, 10

Offset

0,4

Comments

The row sums are zero.

Row n consists of the coefficients in the expansion of n*(x - 1)*(x + 1)^(n - 1). - Franck Maminirina Ramaharo, Oct 02 2018

Links

G. C. Greubel, Table of n, a(n) for n = 0..5150 (Rows n=1..100 of triangle, flattened; offset adapted by Georg Fischer, Jan 31 2019)

Rida T. Farouki, The Bernstein polynomial basis: A centennial retrospective, Computer Aided Geometric Design Vol. 29 (2012), 379-419.

Eric Weisstein's World of Mathematics, Bernstein Polynomial

Wikipedia, Bernstein polynomial

Formula

T(n,k) = n*(B(1/2;n-1,k-1) - B(1/2;n-1,k))*2^(n - 1), where B(t;n,k) = binomial(n,k)*t^k*(1 - t)^(n - k) denotes the k-th Benstein basis polynomial of degree n.

T(n,k) = n*A112467(n,k).

From Franck Maminirina Ramaharo, Oct 02 2018: (Start)

T(n,k) = -T(n,n-k)

T(n,0) = -n.

T(n,1) = -A067998(n)

E.g.f.: (x*y - y)/(x*y + y - 1)^2.

Sum_{k=0..n} abs(T(n,k)) = 2*A100071(n).

Sum_{k=0..n} T(n,k)^2 = 2*A037965(n).

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

Sum_{k=0..n} k^2*T(n,k) = A014477(n-1). (End)

Example

Triangle begins:
    0;
   -1,   1;
   -2,   0,    2;
   -3,  -3,    3,    3;
   -4,  -8,    0,    8,    4;
   -5, -15,  -10,   10,   15,   5;
   -6, -24,  -30,    0,   30,  24,   6;
   -7, -35,  -63,  -35,   35,  63,  35,   7;
   -8, -48, -112, -112,    0, 112, 112,  48,   8;
   -9, -63, -180, -252, -126, 126, 252, 180,  63,  9;
  -10, -80, -270, -480, -420,   0, 420, 480, 270, 80, 10;
  ...

Maple

a:=proc(n, k) n*(binomial(n-1, k-1)-binomial(n-1, k)); end proc: seq(seq(a(n, k), k=0..n), n=0..10); # Muniru A Asiru, Oct 03 2018

Mathematica

Table[Table[n*(Binomial[n - 1, k - 1] - Binomial[n - 1, k]), {k, 0, n}], {n, 0, 12}]//Flatten

Prog

(Maxima) T(n, k) := n*(binomial(n - 1, k - 1) - binomial(n - 1, k))$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$ /* Franck Maminirina Ramaharo, Oct 02 2018 */

Crossrefs

Cf. A007318, A112467, A128433, A128434.

Sequence in context: A307300 A354438 A354470 * A261097 A335335 A261217

Adjacent sequences: A141689 A141690 A141691 * A141693 A141694 A141695

Keyword

easy,tabl,sign

Author

Roger L. Bagula, Sep 09 2008

Extensions

Edited, new name and offset corrected by Franck Maminirina Ramaharo, Oct 02 2018

Status

approved