A065109 Triangle T(n,k) of coefficients relating to Bezier curve continuity.

1, 2, -1, 4, -4, 1, 8, -12, 6, -1, 16, -32, 24, -8, 1, 32, -80, 80, -40, 10, -1, 64, -192, 240, -160, 60, -12, 1, 128, -448, 672, -560, 280, -84, 14, -1, 256, -1024, 1792, -1792, 1120, -448, 112, -16, 1, 512, -2304, 4608, -5376, 4032, -2016, 672, -144, 18, -1, 1024, -5120, 11520, -15360, 13440

Offset: 0

Author

Peter J. Taylor, Nov 12 2001

Keywords

• sign
• tabl
• nice
• easy

Comments

Row sums are 1, antidiagonal sums are the natural numbers. - Gerald McGarvey, May 29 2005

Row sums = 1. - Roger L. Bagula, Sep 12 2008

Riordan array (1/(1-2x), -x/(1-2x)). - Philippe Deléham, Nov 27 2009

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

Examples

			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.
Triangle begins:
     1;
     2,     -1;
     4,     -4,     1;
     8,    -12,     6,     -1;
    16,    -32,    24,     -8,     1;
    32,    -80,    80,    -40,    10,     -1;
    64,   -192,   240,   -160,    60,    -12,     1;
   128,   -448,   672,   -560,   280,    -84,    14,    -1;
   256,  -1024,  1792,  -1792,  1120,   -448,   112,   -16,    1;
   512,  -2304,  4608,  -5376,  4032,  -2016,   672,  -144,   18,   -1;
  1024,  -5120, 11520, -15360, 13440,  -8064,  3360,  -960,  180,  -20,  1;
  2048, -11264, 28160, -42240, 42240, -29568, 14784, -5280, 1320, -220, 22, -1;
		

Links

• Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
• Filippo Disanto, Some Statistics on the Hypercubes of Catalan Permutations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.
• Peter J. Taylor, Conditions for C-a Continuity of Bezier Curves

Crossrefs

Cf. A038207, A013609. Apart from signs, same as A038207.

Programs

Haskell

a065109 n k = a065109_tabl !! n !! k
a065109_row n = a065109_tabl !! n
a065109_tabl = iterate
   (\row -> zipWith (-) (map (* 2) row ++ [0]) ([0] ++ row)) [1]
-- Reinhard Zumkeller, Apr 25 2013

Magma

/* As triangle: */  [[(-1)^k*2^(n-k)*Binomial(n, k): k in [0..n]]: n in [0..15]]; // Vincenzo Librandi, Apr 26 2015

Maple

seq(seq((-1)^k * 2^(n-k) * binomial(n, k), k= 0 .. n), n = 0 .. 12); # Robert Israel, Apr 26 2015

Mathematica

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 *)
Flatten[Table[(-1)^k 2^(n-k) Binomial[n,k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Mar 13 2013 *)

Formula

T(n, k) = (-1)^k * 2^(n-k) * binomial(n, k).

Sum_{i=0..n} binomial(n,i) * (-1)^i * T(i,r) = (-1)^(n-r) * binomial(n,r).

For n > 0, T(n, k) = 2*T(n-1, k) - T(n-1, k-1). - Gerald McGarvey, May 29 2005

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

Sum_{k=0..n} T(n,k)*A000108(k) = A001405(n). - Philippe Deléham, Nov 27 2009

Sum_{k=0..n} T(n,k)*x^k = (2-x)^n. - Philippe Deléham, Dec 15 2009

G.f.: Sum_{n>=0} (2-x)^n * x^(n*(n+1)/2). - Robert Israel, Apr 26 2015

G.f.: 1/(1-2*x+x*y). - R. J. Mathar, Aug 11 2015