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A141692 Triangle read by rows: T(n,k) = n*(binomial(n - 1, k - 1) - binomial(n - 1, k)), 0 <= k <= n. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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, Rows n=1..100 of triangle, flattened

Rita 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: A171731 A185815 A003987 * A261097 A261217 A245230

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

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Last modified October 20 01:39 EDT 2018. Contains 316378 sequences. (Running on oeis4.)