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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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Previous name was: "Expansion of certain polynomials; see formula."
LINKS
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

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)