OFFSET
0,2
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
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).
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)*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
KEYWORD
AUTHOR
Peter J. Taylor, Nov 12 2001
STATUS
approved