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.
-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
Examples
{-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}
Programs
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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
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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]
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).
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
Comments