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A192174 Triangle T(n,k) of the coefficients [x^(n-k)] of the polynomial p(0,x)=-1, p(1,x)=x and p(n,x) = x*p(n-1,x) - p(n-2,x) in row n, column k. 5
-1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, -1, 0, -1, 1, 0, -2, 0, -1, 0, 1, 0, -3, 0, 0, 0, 1, 1, 0, -4, 0, 2, 0, 2, 0, 1, 0, -5, 0, 5, 0, 2, 0, -1, 1, 0, -6, 0, 9, 0, 0, 0, -3, 0, 1, 0, -7, 0, 14, 0, -5, 0, -5, 0, 1, 1, 0, -8, 0, 20, 0, -14, 0, -5, 0, 4, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,18

COMMENTS

Consider the Catalan triangle A009766 antisymmetrically extended by a mirror along the diagonal (see also A176239):

0, -1, -1, -1, -1,  -1,  -1,   -1,

1,  0, -1, -2, -3,  -4,  -5,   -6,

1,  1,  0, -2, -5,  -9, -14,  -20,

1,  2,  2,  0, -5, -14, -28,  -48,

1,  3,  5,  5,  0, -14, -42,  -90,

1,  4,  9, 14, 14,   0, -42, -132,

1,  5, 14, 28, 42,  42,   0, -132,

1,  6, 20, 48, 90, 132, 132,    0.

The rows in this array are essentially the columns of T(n,k).

LINKS

Table of n, a(n) for n=0..77.

FORMULA

Sum_{k=0..n} T(n,k) = A057079(n-1).

Apparently T(3s,2s-2) = (-1)^(s+1)*A000245(s), s >= 1.

EXAMPLE

Triangle begins

  -1;      # -1

   1,  0;      # x

   1,  0,  1;      # x^2+1

   1,  0,  0,  0;      # x^3

   1,  0, -1,  0, -1;      # x^4-x^2-1

   1,  0, -2,  0, -1,  0;

   1,  0, -3,  0,  0,  0,  1;

   1,  0, -4,  0,  2,  0,  2,  0;

   1,  0, -5,  0,  5,  0,  2,  0, -1;

   1,  0, -6,  0,  9,  0,  0,  0, -3,  0;

   1,  0, -7,  0, 14,  0, -5,  0, -5,  0,  1;

   1,  0, -8,  0, 20,  0,-14,  0, -5,  0,  4,  0;

   1,  0, -9,  0, 27,  0,-28,  0,  0,  0,  9,  0, -1;

MAPLE

p:= proc(n, x) option remember: if n=0 then -1 elif n=1 then x elif n>=2 then x*procname(n-1, x)-procname(n-2, x) fi: end: A192174 := proc(n, k): coeff(p(n, x), x, n-k): end: seq(seq(A192174(n, k), k=0..n), n=0..11); # Johannes W. Meijer, Aug 21 2011

CROSSREFS

Cf. A023443, A080956 and A000096, A129936 and A005586, A005587.

Cf. A194084. - Paul Curtz, Aug 16 2011

Sequence in context: A035169 A275851 A067432 * A262202 A284413 A323879

Adjacent sequences:  A192171 A192172 A192173 * A192175 A192176 A192177

KEYWORD

sign,tabl,easy

AUTHOR

Paul Curtz, Jun 24 2011

STATUS

approved

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Last modified July 9 16:50 EDT 2020. Contains 335545 sequences. (Running on oeis4.)