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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
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
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
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