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A065109
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Triangle T(n,k) of coefficients relating to Bezier curve continuity.
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3
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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; internal format)
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OFFSET
| 0,2
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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 (gerald.mcgarvey(AT)comcast.net), May 29 2005
Row sums = one. - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Sep 12 2008
Riordan array (1/(1-2x), -x/(1-2x)). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 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 DELEHAM (kolotoko(AT)wanadoo.fr), Dec 15 2009]
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LINKS
| Peter J. Taylor, Conditions for C-a Continuity of Bezier Curves
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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 (gerald.mcgarvey(AT)comcast.net), 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 (rlbagulatftn(AT)yahoo.com), Sep 12 2008
Sum_{k, 0<=k<=n} T(n,k)*A000108(k)= A001405(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 27 2009]
Sum_{k 0<=k<=n} T(n,k)*x^k = (2-x)^n. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 15 2009]
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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}
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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 (rlbagulatftn(AT)yahoo.com), Sep 12 2008
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CROSSREFS
| Cf. A038207, A013609. Apart from signs, same as A038207.
Sequence in context: A134397 A134395 A038207 * A113988 A134308 A202710
Adjacent sequences: A065106 A065107 A065108 * A065110 A065111 A065112
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KEYWORD
| easy,sign,tabl,nice
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AUTHOR
| Peter J. Taylor, Nov 12 2001
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