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A065109 Triangle T(n,k) of coefficients relating to Bezier curve continuity. 9
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Sum(binomial(n,i) * (-1)^i * T(i,r), i=0..n) = (-1)^(n-r) * binomial(n,r).

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

Row sums = one. - 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. [From Philippe Deléham, Dec 15 2009]

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

FORMULA

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

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[(-1)^m*Multinomial[n - m - k, m, k],{k,0,n}]. - Roger L. Bagula, Sep 12 2008

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

Sum_{k 0<=k<=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

EXAMPLE

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}

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A134397 A134395 A038207 * A113988 A134308 A209240

Adjacent sequences:  A065106 A065107 A065108 * A065110 A065111 A065112

KEYWORD

sign,tabl,nice,easy

AUTHOR

Peter J. Taylor, Nov 12 2001

STATUS

approved

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Last modified October 20 14:20 EDT 2018. Contains 316380 sequences. (Running on oeis4.)