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%I #44 Feb 17 2022 10:00:49
%S 1,2,-1,4,-4,1,8,-12,6,-1,16,-32,24,-8,1,32,-80,80,-40,10,-1,64,-192,
%T 240,-160,60,-12,1,128,-448,672,-560,280,-84,14,-1,256,-1024,1792,
%U -1792,1120,-448,112,-16,1,512,-2304,4608,-5376,4032,-2016,672,-144,18,-1,1024,-5120,11520,-15360,13440
%N Triangle T(n,k) of coefficients relating to Bezier curve continuity.
%C Row sums are 1, antidiagonal sums are the natural numbers. - _Gerald McGarvey_, May 29 2005
%C Row sums = 1. - _Roger L. Bagula_, Sep 12 2008
%C Riordan array (1/(1-2x), -x/(1-2x)). - _Philippe Deléham_, Nov 27 2009
%C Triangle T(n,k), read by rows, given by [2,0,0,0,0,0,0,0,...] DELTA [ -1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Dec 15 2009
%H Reinhard Zumkeller, <a href="/A065109/b065109.txt">Rows n = 0..120 of triangle, flattened</a>
%H Filippo Disanto, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Disanto/disanto5.html">Some Statistics on the Hypercubes of Catalan Permutations</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.
%H Peter J. Taylor, <a href="http://cheddarmonk.org/papers/BezierContinuity.pdf">Conditions for C-a Continuity of Bezier Curves</a>
%F T(n, k) = (-1)^k * 2^(n-k) * binomial(n, k).
%F Sum_{i=0..n} binomial(n,i) * (-1)^i * T(i,r) = (-1)^(n-r) * binomial(n,r).
%F For n > 0, T(n, k) = 2*T(n-1, k) - T(n-1, k-1). - _Gerald McGarvey_, May 29 2005
%F p(n,m,k) = (-1)^m*multinomial(n - m - k, m, k); t(n,m) = Sum_{k=0..n} (-1)^m*multinomial(n - m - k, m, k). - _Roger L. Bagula_, Sep 12 2008
%F Sum_{k=0..n} T(n,k)*A000108(k) = A001405(n). - _Philippe Deléham_, Nov 27 2009
%F Sum_{k=0..n} T(n,k)*x^k = (2-x)^n. - _Philippe Deléham_, Dec 15 2009
%F G.f.: Sum_{n>=0} (2-x)^n * x^(n*(n+1)/2). - _Robert Israel_, Apr 26 2015
%F G.f.: 1/(1-2*x+x*y). - _R. J. Mathar_, Aug 11 2015
%e For C-2 continuity between P and Q we require Q_0 = P_n; Q_1 = 2P_n - P_n-1; Q_2 = 4P_n - 4P_n-1 + P_n-2.
%e Triangle begins:
%e 1;
%e 2, -1;
%e 4, -4, 1;
%e 8, -12, 6, -1;
%e 16, -32, 24, -8, 1;
%e 32, -80, 80, -40, 10, -1;
%e 64, -192, 240, -160, 60, -12, 1;
%e 128, -448, 672, -560, 280, -84, 14, -1;
%e 256, -1024, 1792, -1792, 1120, -448, 112, -16, 1;
%e 512, -2304, 4608, -5376, 4032, -2016, 672, -144, 18, -1;
%e 1024, -5120, 11520, -15360, 13440, -8064, 3360, -960, 180, -20, 1;
%e 2048, -11264, 28160, -42240, 42240, -29568, 14784, -5280, 1320, -220, 22, -1;
%p seq(seq((-1)^k * 2^(n-k) * binomial(n, k), k= 0 .. n), n = 0 .. 12); # _Robert Israel_, Apr 26 2015
%t t[n_, m_, k_] = (-1)^m*Multinomial[n - m - k, m, k]; Table[Table[Sum[t[n, m, k], {k, 0, n}], {m, 0, n}], {n, 0, 11}]; Flatten[%] (* _Roger L. Bagula_, Sep 12 2008 *)
%t Flatten[Table[(-1)^k 2^(n-k) Binomial[n,k],{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, Mar 13 2013 *)
%o (Haskell)
%o a065109 n k = a065109_tabl !! n !! k
%o a065109_row n = a065109_tabl !! n
%o a065109_tabl = iterate
%o (\row -> zipWith (-) (map (* 2) row ++ [0]) ([0] ++ row)) [1]
%o -- _Reinhard Zumkeller_, Apr 25 2013
%o (Magma) /* As triangle: */ [[(-1)^k*2^(n-k)*Binomial(n, k): k in [0..n]]: n in [0..15]]; // _Vincenzo Librandi_, Apr 26 2015
%Y Cf. A038207, A013609. Apart from signs, same as A038207.
%K sign,tabl,nice,easy
%O 0,2
%A _Peter J. Taylor_, Nov 12 2001
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