A137289 - OEIS

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

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OFFSET

1,2

COMMENTS

Previous name was: "Expansion of certain polynomials; see formula."

LINKS

Table of n, a(n) for n=1..66.

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

Cf. A135929, A138034.

Sequence in context: A333310 A049986 A218797 * A211359 A211357 A238416

Adjacent sequences: A137286 A137287 A137288 * A137290 A137291 A137292

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