A097418 Triangle of coefficients of a certain sequence of polynomials f_n(x) arising in connection with deformations of coordinate rings of type D Kleinian singularities.

1, 2, 0, 3, 2, 0, 4, 8, 8, 0, 5, 20, 56, 56, 0, 6, 40, 216, 608, 608, 0, 7, 70, 616, 3352, 9440, 9440, 0, 8, 112, 1456, 12928, 70400, 198272, 198272, 0, 9, 168, 3024, 39696, 352768, 1921152, 5410688, 5410688, 0, 10, 240, 5712, 103872, 1364800, 12129664, 66057856, 186043904, 186043904, 0

Offset

1,2

Comments

f_n(x) has the property that whenever (a,b) is a pair of complex numbers satisfying 2ab = a^2 + 2a + b^2 + 2b, we have f_n(a) + f_n(b) = 2(a^n - b^n)/(a-b) (interpreted as 2na^(n-1) if a=b). Using the pairs (0,0), (0,-2), (-2,-6), (-6,-12), (-12,-20), ...(see A002378), this enables us to successively deduce the values of f_n(0), f_n(-2),... (and this of course determines f_n(x)). There may be no simpler characterization.

Links

Table of n, a(n) for n=1..55.

Paul Boddington, No-cycle algebras and representation theory, Ph.D. thesis, University of Warwick, 2004.

Example

The array begins
  1
  2 0
  3 2 0
  4 8 8 0
corresponding to the polynomials f_1(x) = 1, f_2(x) = 2x, f_3(x) = 3x^2 + 2x, f_4(x) = 4x^3 + 8x^2 + 8x.

Prog

(PARI) f(n, a, b) = if (a==b, 2*n*a^(n-1), 2*(a^n - b^n)/(a-b));
row(n) = if (n==1, return([1])); my(v = vector(n-1, k, f(n, -(k-1)*k, -k*(k+1)))); my(m = matrix(n-1, n-1, i, j, (-j*(j-1))^i + (-(j+1)*j)^i)); concat(Vecrev(v/m), 0); \\ Michel Marcus, Mar 19 2023

Crossrefs

Cf. A002378, A005439 (right diagonal).

Sequence in context: A269133 A143324 A287416 * A362789 A154752 A271868

Adjacent sequences: A097415 A097416 A097417 * A097419 A097420 A097421

Keyword

nonn,tabl

Author

Paul Boddington, Aug 20 2004

Extensions

Typo corrected and extended by Michel Marcus, Mar 19 2023

Status

approved