A181472 Riordan array ((1+x)/(1+2*x+2*x^2),x*(1+x)/(1+2*x+2*x^2)).

1, -1, 1, 0, -2, 1, 2, 1, -3, 1, -4, 4, 3, -4, 1, 4, -12, 5, 6, -5, 1, 0, 16, -24, 4, 10, -6, 1, -8, -4, 42, -39, 0, 15, -7, 1, 16, -32, -24, 88, -55, -8, 21, -8, 1, -16, 80, -72, -80, 159, -69, -21, 28, -9, 1, 0, -96, 240, -112, -200, 258, -77, -40, 36, -10, 1

Offset

0,5

Comments

Inverse is A054336. Coefficient array for Faber polynomials (of second kind) defined by f(x) = x+1-Sum_{k>=1} (-1)^k/x^k.

Subtriangle of the triangle given by (0, -1, 1, -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, Feb 20 2013

Links

Table of n, a(n) for n=0..65.

Formula

T(n,m) = Sum_{k=m..n} (-2)^(k-m)*binomial(k,n-k)*binomial(k-1,m-1), n,m>0. - Vladimir Kruchinin, Mar 09 2011

T(n,k) = T(n-1,k-1) + T(n-2,k-1) -2*T(n-1,k) - 2*T(n-2,k), T(0,0) = T(1,1) = 1, T(1,0) = -1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Feb 20 2013

G.f.: (-1-x)/(-1-2*x-2*x^2+x*y+x^2*y). - R. J. Mathar, Aug 12 2015

Example

Triangle begins:
  1,
  -1, 1,
  0, -2, 1,
  2, 1, -3, 1,
  -4, 4, 3, -4, 1,
  4, -12, 5, 6, -5, 1,
  0, 16, -24, 4, 10, -6, 1,
  -8, -4, 42, -39, 0, 15, -7, 1,
  16, -32, -24, 88, -55, -8, 21, -8, 1
Production matrix is
  -1, 1,
  -1, -1, 1,
  0, -1, -1, 1,
  -1, 0, -1, -1, 1,
  0, -1, 0, -1, -1, 1,
  -2, 0, -1, 0, -1, -1, 1,
  0, -2, 0, -1, 0, -1, -1, 1,
  -5, 0, -2, 0, -1, 0, -1, -1, 1,
  0, -5, 0, -2, 0, -1, 0, -1, -1, 1
  -14, 0, -5, 0, -2, 0, -1, 0, -1, -1, 1
based on the aerated Catalan numbers.
Triangle (0, -1, 1, -2, 0, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
  1
  0, 1
  0, -1, 1
  0, 0, -2, 1
  0, 2, 1, -3, 1
  0, -4, 4, 3, -4, 1
  0, 4, -12, 5, 6, -5, 1. - Philippe Deléham, Feb 20 2013

Crossrefs

Sequence in context: A153174 A138222 A138224 * A271825 A271875 A324194

Adjacent sequences: A181469 A181470 A181471 * A181473 A181474 A181475

Keyword

easy,sign,tabl

Author

Paul Barry, Oct 21 2010

Status

approved