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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.

%I A141692 #30 Feb 16 2025 08:33:08
%S A141692 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,
%T A141692 30,24,6,-7,-35,-63,-35,35,63,35,7,-8,-48,-112,-112,0,112,112,48,8,-9,
%U A141692 -63,-180,-252,-126,126,252,180,63,9,-10,-80,-270,-480,-420,0,420,480,270,80,10
%N A141692 Triangle read by rows: T(n,k) = n*(binomial(n - 1, k - 1) - binomial(n - 1, k)), 0 <= k <= n.
%C A141692 The row sums are zero.
%C A141692 Row n consists of the coefficients in the expansion of n*(x - 1)*(x + 1)^(n - 1). - _Franck Maminirina Ramaharo_, Oct 02 2018
%H A141692 G. C. Greubel, <a href="/A141692/b141692.txt">Table of n, a(n) for n = 0..5150</a> (Rows n=1..100 of triangle, flattened; offset adapted by _Georg Fischer_, Jan 31 2019)
%H A141692 Rida T. Farouki, <a href="https://doi.org/10.1016/j.cagd.2012.03.001">The Bernstein polynomial basis: A centennial retrospective</a>, Computer Aided Geometric Design Vol. 29 (2012), 379-419.
%H A141692 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BernsteinPolynomial.html">Bernstein Polynomial</a>
%H A141692 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bernstein_polynomial">Bernstein polynomial</a>
%F A141692 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.
%F A141692 T(n,k) = n*A112467(n,k).
%F A141692 From _Franck Maminirina Ramaharo_, Oct 02 2018: (Start)
%F A141692 T(n,k) = -T(n,n-k)
%F A141692 T(n,0) = -n.
%F A141692 T(n,1) = -A067998(n)
%F A141692 E.g.f.: (x*y - y)/(x*y + y - 1)^2.
%F A141692 Sum_{k=0..n} abs(T(n,k)) = 2*A100071(n).
%F A141692 Sum_{k=0..n} T(n,k)^2 = 2*A037965(n).
%F A141692 Sum_{k=0..n} k*T(n,k) = A001787(n).
%F A141692 Sum_{k=0..n} k^2*T(n,k) = A014477(n-1). (End)
%e A141692 Triangle begins:
%e A141692     0;
%e A141692    -1,   1;
%e A141692    -2,   0,    2;
%e A141692    -3,  -3,    3,    3;
%e A141692    -4,  -8,    0,    8,    4;
%e A141692    -5, -15,  -10,   10,   15,   5;
%e A141692    -6, -24,  -30,    0,   30,  24,   6;
%e A141692    -7, -35,  -63,  -35,   35,  63,  35,   7;
%e A141692    -8, -48, -112, -112,    0, 112, 112,  48,   8;
%e A141692    -9, -63, -180, -252, -126, 126, 252, 180,  63,  9;
%e A141692   -10, -80, -270, -480, -420,   0, 420, 480, 270, 80, 10;
%e A141692   ...
%p A141692 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
%t A141692 Table[Table[n*(Binomial[n - 1, k - 1] - Binomial[n - 1, k]),{k, 0, n}],{n, 0, 12}]//Flatten
%o A141692 (Maxima) T(n, k) := n*(binomial(n - 1, k - 1) - binomial(n - 1, k))$
%o A141692 tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$ /* _Franck Maminirina Ramaharo_, Oct 02 2018 */
%Y A141692 Cf. A007318, A112467, A128433, A128434.
%K A141692 easy,tabl,sign
%O A141692 0,4
%A A141692 _Roger L. Bagula_, Sep 09 2008
%E A141692 Edited, new name and offset corrected by _Franck Maminirina Ramaharo_, Oct 02 2018