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A137289
Triangle read by rows: T(n,k) = (-1)^(n-k)*(C(k+n-1,n-k)-2*C(k+n-1,n-k-1)) for n>=0 and 0<=k<=n.
2
-1, 2, 1, -2, 0, 1, 2, -3, -2, 1, -2, 8, 0, -4, 1, 2, -15, 10, 7, -6, 1, -2, 24, -35, 0, 18, -8, 1, 2, -35, 84, -42, -30, 33, -10, 1, -2, 48, -168, 168, 0, -88, 52, -12, 1, 2, -63, 300, -462, 198, 143, -182, 75, -14, 1, -2, 80, -495, 1056, -858, 0, 455, -320, 102, -16, 1
OFFSET
1,2
COMMENTS
Previous name was: "Expansion of certain polynomials; see formula."
FORMULA
B(x, 0) = -1, B(x, 2) = x^2 + 2, B(x, 3) = x^3 + x, B(x, 4) = x^4 - 2, and B(x, n) = x*B(x, n - 1) - B(x, n - 2) for n>=2, expand B(sqrt(x), 2*n).
EXAMPLE
{-1},
{2, 1},
{-2, 0, 1},
{2, -3, -2, 1},
{-2, 8, 0, -4, 1},
{2, -15, 10, 7, -6, 1},
{-2, 24, -35, 0, 18, -8, 1},
{2, -35, 84, -42, -30, 33, -10, 1},
{-2, 48, -168, 168,0, -88, 52, -12, 1},
{2, -63, 300, -462, 198, 143, -182, 75, -14,1},
{-2, 80, -495, 1056, -858, 0, 455, -320, 102, -16, 1}
MAPLE
T := (n, k) -> (-1)^(n-k)*(binomial(k+n-1, n-k)-2*binomial(k+n-1, n-k-1)):
seq(seq(T(n, k), k=0..n), n=0..10); # Peter Luschny, May 15 2016
MATHEMATICA
B[x, 0] = -1; B[x, 1] = x; B[x, 2] = 2 + x^2; B[x, 4] = -2 + x^4; B[ x, 3] = x + x^3; B[x_, n_] := B[x, n] = x*B[x, n - 1] - B[x, n - 2]; a = Table[CoefficientList[B[x, n] /. x -> Sqrt[y], y], {n, 0, 20, 2}]; Flatten[a]
CROSSREFS
Sequence in context: A333310 A049986 A218797 * A211359 A211357 A238416
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Mar 14 2008
EXTENSIONS
Edited by N. J. A. Sloane, Jan 05 2009
Edited by Joerg Arndt, Nov 15 2014
New name and changed a(1) to -1 by Peter Luschny, May 15 2016
STATUS
approved