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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
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 (list; table; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,tabl
AUTHOR
Paul Boddington, Aug 20 2004
EXTENSIONS
Typo corrected and extended by Michel Marcus, Mar 19 2023
STATUS
approved

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Last modified March 2 14:21 EST 2024. Contains 370493 sequences. (Running on oeis4.)