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A132460 Triangle read by rows of the initial floor(n/2) + 1 coefficients of 1/C(x)^n, where C(x) is the g.f. of the Catalan sequence (A000108). 4
1, 1, 1, -2, 1, -3, 1, -4, 2, 1, -5, 5, 1, -6, 9, -2, 1, -7, 14, -7, 1, -8, 20, -16, 2, 1, -9, 27, -30, 9, 1, -10, 35, -50, 25, -2, 1, -11, 44, -77, 55, -11, 1, -12, 54, -112, 105, -36, 2, 1, -13, 65, -156, 182, -91, 13, 1, -14, 77, -210, 294, -196, 49, -2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Essentially equals a signed version of A034807, the triangle of Lucas polynomials. The initial n coefficients of 1/C(x)^n consist of row n followed by floor((n-1)/2) zeros for n > 0.

For the following formula for 1/C(x)^n see the W. Lang reference, proposition 1 on p. 411:

1/C(x)^n = (sqrt(x))^n*(S(n,1/sqrt(x)) - sqrt(x)*S(n-1,1/sqrt(x))*C(x)), n >= 0, with the Chebyshev polynomials S(n,x) with coefficients given in A049310. See also the coefficient array A115139 for P(n,x) = (sqrt(x)^(n-1))*S(n-1, 1/sqrt(x))). - Wolfdieter Lang, Sep 14 2013

This triangular array is composed of interleaved rows of reversed, A127677 (cf. A156308, A217476, A263916) and reversed, signed A111125. - Tom Copeland, Nov 07 2015

It seems that the n-th row lists the coefficients of the HOMFLYPT (HOMFLY) polynomial reduced to one variable for link family n, see Jablan's slide 38. - Andrey Zabolotskiy, Jan 16 2018

LINKS

Table of n, a(n) for n=0..63.

T. Copeland, Addendum to Elliptic Lie Triad

G. Dattoli, E. Di Palma, E. Sabia, Cardan Polynomials, Chebyshev Exponents, Ultra-Radicals and Generalized Imaginary Units, Advances in Applied Clifford Algebras, 2014.

Pentti Haukkanen, Jorma Merikoski, Seppo Mustonen, Some polynomials associated with regular polygons, Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 178-193.

S. Jablan, Knots, computers, conjectures

Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eq. (23) with n -> -n and eq. (20).

FORMULA

T(n,k) = (-1)^k*( C(n-k,k) + C(n-k-1,k-1) ) for n >= 0, 0 <= k <= floor(n/2).

EXAMPLE

Triangle begins:

  1;

  1;

  1,   -2;

  1,   -3;

  1,   -4,    2;

  1,   -5,    5;

  1,   -6,    9,   -2;

  1,   -7,   14,   -7;

  1,   -8,   20,  -16,    2;

  1,   -9,   27,  -30,    9;

  1,  -10,   35,  -50,   25,   -2;

  1,  -11,   44,  -77,   55,  -11;

  1,  -12,   54, -112,  105,  -36,    2;

  1,  -13,   65, -156,  182,  -91,   13;

  1,  -14,   77, -210,  294, -196,   49,   -2; ...

PROG

(PARI) {T(n, k)=if(k>n\2, 0, (-1)^k*(binomial(n-k, k)+binomial(n-k-1, k-1)))}

CROSSREFS

Cf. A132461 (row squared sums); A034807 (Lucas polynomials); A000108.

Cf. A111125, A127677, A156308, A217476, A263916.

Sequence in context: A120385 A216477 A195836 * A238800 A067734 A161904

Adjacent sequences:  A132457 A132458 A132459 * A132461 A132462 A132463

KEYWORD

sign,tabf

AUTHOR

Paul D. Hanna, Aug 21 2007

STATUS

approved

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Last modified February 20 15:30 EST 2018. Contains 299380 sequences. (Running on oeis4.)