0,2

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

Peter J. Taylor, Conditions for C-a Continuity of Bezier Curves

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

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}

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

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

Sequence in context: A134397 A134395 A038207 this_sequence A113988 A134308 A117427

Adjacent sequences: A065106 A065107 A065108 this_sequence A065110 A065111 A065112

easy,sign,tabl,nice

Peter J. Taylor, Nov 12 2001