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

Author

Paul Boddington, Aug 20 2004

Keywords

• nonn
• tabl

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A002378, A005439 (right diagonal).

Programs

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

Extensions

Typo corrected and extended by Michel Marcus, Mar 19 2023